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c^O  I 

Loud on- 


STATL 

,  -^  Southern  Branch 

of  the 

University  of  California 

Los  Angeles 

Form  Ll 


JONES' 

BOOK  STORE, 


This  book  is  DUE  on  the  last  date  stamped  below 


1  1 1924 


V 


1     192, 


A    LABORATORY   COURSE 


EXPERIMENTAL   PHYSICS 


A   LABORATORY  COURSE 


EXPERIMENTAL  PHYSICS 


r,\ 


W.   J.    LOUDON,    B.A. 

DEMONSTRATOR   IN   PHYSICS  IN  THE  UNIVERSITY  OF  TORONTO 
AND 

J.    C.    McLENNAN,    B.A. 

ASSISTANT  DEMONSTRATOR  IN  PHYSICS  IN  THE  UNIVERSITY   OF  TORONTO 


MACMILLAN    AND    CO. 

AND    LONDON 

I895 
All  rights  reserved 


COPYRIGHT,  1895, 
MACMILLAN  AND  CO. 


NortoooH  $rrss : 

J.  S.  Gushing  &  Co.  — Berwick  &  Smith. 
Norwood,  Mass.,  U.S.A. 


PREFACE. 


AT  the  present  day,  when  students  are  required  to  gain 
knowledge  of  natural  phenomena  by  performing  experiments 
for  themselves  in  laboratories,  every  teacher  finds  that  as  his 
classes  increase  in  number,  some  difficulty  is  experienced  in 
providing,  during  a  limited  time,  ample  instruction  in  the 
matter  of  details  and  methods. 

During  the  past  few  years  we  ourselves  have  had  such  diffi- 
culties with  large  classes ;  and  that  is  our  reason  for  the 
appearance  of  the  present  work,  which  is  the  natural  out- 
come of  our  experience.  We  know  that  it  will  be  of  service 
to  our  own  students,  and  hope  that  it  will  be  appreciated  by 
those  engaged  in  teaching  Experimental  Physics  elsewhere. 

The  book  contains  a  series  of  elementary  experiments  speci- 
ally adapted  for  students  who  have  had  but  little  acquaintance 
with  higher  mathematical  methods  :  these  are  arranged  as  far 
as  possible  in  order  of  difficulty.  There  is  also  an  advanced 
course  of  experimental  work  in  Acoustics,  Heat,  and  Electricity 
and  Magnetism,  which  is  intended  for  those  who  have  taken 
the  elementary  course. 

The  experiments  in  Acoustics  are  simple,  and  of  such  a 
nature  that  the  most  of  them  can  be  performed  by  beginners 
in  the  study  of  Physics ;  those  in  Heat,  although  not  requiring 
more  than  an  ordinary  acquaintance  with  Arithmetic,  are  more 
tedious  and  apt  to  test  the  patience  of  the  experimenter  ;  while 


vi  PREFACE. 

the  course  in  Electricity  and  Magnetism  has  been  arranged  to 
illustrate  the  fundamental  laws  of  the  mathematical  theory,  and 
involves  a  good  working  knowledge  of  the  Calculus. 

The  important  subject  of  Physical  Optics  has  been  omitted 
because  we  have  been  unable  to  map  out  a  course  of  experi- 
ments which  would  be  in  accordance  with  the  ordinarily 
accepted  theory  of  Fresnel.  At  an  early  date,  however,  we 
hope  to  place  before  the  public  a  connected  series  of  experi- 
ments (based  upon  true  fundamental  laws)  in  that  most 
interesting  of  all  physical  studies. 

A  short  appendix  is  given  on  the  methods  of  determining 
the  value  of  Gravity  and  on  the  use  of  the  Torsion  Pendulum. 

Finally,  we  may  say  that  throughout  the  work  simplifica- 
tion of  method  has  been  our  greatest,  if  not  our  only,  aim  ; 
for  therein  we  believe  lies  the  only  true  sign  of  Progress. 

UNIVERSITY  OF  TORONTO,  April,  1895. 


LIST   OF  EXPERIMENTS    IN    THE   ELE- 
MENTARY  COURSE. 


PAGE 

I.   The  Vernier 3 

II.   The  Calipers               .     .  > 6 

III.  The  Cathetometer 8 

IV.  The  Spherometer 10 

V.    Micrometer  Screw  Gauge  ...                  ...  12 

VI.    Dividing  Engine        ....                  ...  13 

VII.    Specific  Gravity  Bottle      ....                  .         .  15 

VIII.   Hydrostatic  Balance 16 

IX.    Nicholson's  Hydrometer    .......  19 

X.    Fahrenheit's  Hydrometer  .......  20 

XI.    Mohr's  Balance 22 

XII.    Boyle's  Law 24 

(A)  For  pressures  greater  than  an  atmosphere. 
(.#)  For  pressures  less  than  an  atmosphere. 

XIII.  The  Volumenometer           .......  26 

XIV.  Determination  of  Capillary  Constants        ....  28 
XV.    The  Sextant 33 

XVI.   The  Goniometer.  —  Two  Methods     .....  36 

XVII.    Radius  of  Curvature  of  a  Concave  Mirror         ...  39 

XVIII.    Radius  of  Curvature  of  a  Convex  Mirror          ...  41 
XIX.    Radius  of  Curvature  of  a  Spherical  Reflecting  Surface.  — 

General  Method          •    ,    ••        •                  •         -  44 

XX.    Focal  Length  of  a  Biconvex  Lens.  —  Two  Methods         .  47 
vii 


viii  LIST   OF   EXPERIMENTS. 

PAGE 

XXI.  Focal  Length  of  a  Biconcave  Lens.  —  Two  Methods        .  51 

XXII.  Indices  of  Refraction          .         .         .                           .         .  54 

XXIII.  Examples  of  Magnification  with  Lenses    ....  58 

XXIV.  Exercises  with  Photographic  Lenses          ....  64 
XXV.  Bunsen's  Photometer 68 

XXVI.  Ayrton's  Photometer.  —  Modified  from  Bunsen's       .         .  70 

XXVII.  Rotating  Disc  Photometer         ......  73 

XXVIII.  Specific  Heat  of  Solids     .......  76 

XXIX.  Specific  Heat  of  Liquids  ........  80 

XXX.  Latent  Heat  of  Fusion      .                 83 

XXXI.  Level  Testing    .         .        .  '     .                 ....  85 


PART    I. 
ELEMENTARY   COURSE 


A  LABORATORY  COURSE  IN   EXPERI- 
MENTAL  PHYSICS. 


EXPERIMENTS. 

I.     THE   VERNIER. 

To  measuring  instruments  provided  with  a  scale  there  is 
frequently  attached  a  more  finely  divided  one,  by  means  of  which 
readings  can  be  made  with  greater  precision.  This  latter  scale 
is  called  a  vernier,  and  is  arranged  so  as  to  slide  along  the  scale 
proper,  either  edge  to  edge,  or  else  one  overlapping  the  other. 
In  this  latter  case  the  edge  of  the  upper  scale  is  beveled. 
Generally  a  length  equal  to  n  —  I  divisions  on  the  scale  is  taken 
on  the  vernier,  and  this  length  divided  into  n  equal  parts,  so 
that  one  of  the  vernier  divisions  is  equal  to  «  —  i/n  of  a  scale 
division.  In  this  way,  if  the  zero  line  on  the  vernier  coincides 
with  a  line  on  the  scale,  then  the  space  between  line  I  on  the 
vernier,  and  the  line  next  to  it  on  the  scale  is  i/n  of  a  scale 
division,  that  between  line  2  on  the  vernier,  and  the  line  next 
to  it  on  the  scale  is  equal  to  2/n  of  a  scale  division,  while  that 
between  the  rth  line  on  the  vernier,  and  the  line  next  to  it  on 
the  scale  is  equal  to  r/n  of  a  scale  division,  and  the  ;/th  line  on 
the  vernier  will,  just  as  the  zero  line,  coincide  with  a  line  on  the 

3 


EXPERIMENTAL   PHYSICS. 


scale.  By  following  out  a  process,  the  converse  of  that  just 
described,  it  will  be  readily  seen  that  if  the  rth  line  on  the 
vernier  coincides  with  a  line  on  the  scale,  then  the  space 
between  the  vernier  zero,  and  the  line  next  behind  it  on  the 
scale  is  rjn  of  a  scale  division. 


I 

1  1  1  i  1  1  1 

1  1  1 

1 

1  1 

II  1  1  1  1  1  \ 

1  1  1 

1 

1  1 

Fig.  1. 


Different  values  are  given  to  »,  according  to  the  purpose  for 
which  the  vernier  is  to  be  used.  Figure  i  is  an  example  of  a 
vernier  in  which  n  is  equal  to  10.  In  this  case,  line  5  on  the 
vernier  coincides  with  a  line  on  the  scale,  so  that  the  space 
between  the  vernier  zero,  and  the  line  on 
the  scale  next  behind  it  is  T5^  of  a  scale 
division.  The  reading  indicated  is  there- 
fore 6.5. 

In  the  vernier  on  the  right-hand  side 
of  Fig.  2,  n  is  taken  equal  to  25.  Line 
18  on  the  vernier  coincides  with  a  line  on 
the  scale,  and  the  vernier  zero  is  therefore 
|-|  of  a  scale  division  above  the  line  next 
behind  it  on  the  scale.  Here  one  of  the 
small  scale  divisions  is  equal  to  .05  of  a 
large  one,  and  therefore  the  space  men- 
tioned above  will  be  equal  to  .036  of  a 
large  scale  division,  and  the  reading  is 

FIP*    2 

30.05  +  .036,  or  30.086. 

In  Fig.  3,  A,  J3,  C,  and  D  are  examples  of  this  class  of  ver- 
nier, and  it  will  be  an  instructive  exercise  for  the  student  to 
verify  the  readings  there  given.  In  order  to  avoid  the  calcula- 
tion just  indicated,  verniers  are  often  so  numbered  that  readings 


THE   VERNIER. 


can  be  made  directly.  One  of  this  class  is  exhibited  on  the  left- 
hand  side  of  Fig.  2.  There  n  is  equal  to  20,  and  since  line  19 
on  the  vernier  coincides  with  a  line  on  the  scale  there  is  there- 


fore £$,  or  .95,  of  a  scale  division  between  the  vernier  zero  and 
the  line  next  behind  it  on  the  scale.  From  the  manner  in  which 
the  spaces  are  numbered,  it  can  be  readily  seen  that  line  19  is 


Fig.  4. 


represented  by  95,  and  the  reading  76.395  may  thus  be  made 
directly.  It  frequently  happens,  when  a  small  value  is  given 
to  n,  that  no  line  on  the  vernier  coincides  with  one  on  the  scale. 


6  EXPERIMENTAL   PHYSICS. 

An  example  of  this  is  shewn  in  E  of  Fig.  3,  where  an  approxi- 
mation has  been  made. 

Figure  4  is  an  example  of  a  circular  vernier.  In  this  case,  n 
is  equal  to  30,  and  since  the  small  scale  divisions  are  each  equal 
to  one-half  a  degree  readings  can  be  taken  to  minutes.  The 
reading  indicated  is  13°  37'.  Besides  those  already  described, 
verniers  are  sometimes  constructed  in  which  n  —  \  of  their 
divisions  are  equal  to  n  scale  divisions.  However,  in  this 
case  no  special  difficulty  will  be  met  with,  as  the  same  prin- 
ciples apply  to  all  classes  of  verniers. 


II.     THE    CALIPERS. 

This  instrument,  shewn  in  Fig.  5,  is  essentially  a  graduated 
metal  scale,  to  one  end  of  which  (and  perpendicular  to  it)  is 
attached  a  bar  EP.  Another  bar,  FQ,  of  the  same  form,  is  capa- 


Fig.  5. 

ble  of  being  moved  along  AB,  and  carries  with  it  a  slider  CD, 
which  can  be  fixed  by  the  screw  V  at  any  point  on  AB.  There 
is  a  small  rectangular  opening  in  CD,  and  on  one  of  its  bevelled 
edges  a  vernier  is  ruled,  which  is  so  placed  that  when  the  bars 
EP  and  FQ  are  together,  the  zero  on  the  vernier  corresponds 


THE   CALIPERS.  7 

with  the  zero  on  the  scale.  In  measuring  the  length  of  an 
object,  place  it  between  the  branches  PA  and  QC,  and  gently 
press  the  slider  against  it.  Then  note  the  reading  on  the 
vernier. 

The  calipers  may  also  be  used  to  measure  the  width  of  an 
opening  in  a  tube,  or  other  object,  provided  it  is  greater  than 
the  total  thickness  of  EP  and  FQ  when  they  are  together.  In 
this  case,  insert  AE  and  CF  in  the  opening,  and  separate  them 
as  far  as  its  walls  will  permit.  To  the  vernier  reading  then 
must  be  added  the  thickness  of  AE  and  CF  when  they  are 


Fig.  6. 

together,  which  may  be  found  by  means  of  another  instrument. 
It  will  be  a  useful  exercise  for  the  student  to  test,  at  various 
times,  the  accuracy  of  the  calipers,  by  means  of  a  standard 
instrument,  one  of  which  is  shewn  in  Fig.  6. 

In  this  standard  the  object  to  be  measured  is  placed  between 
the  two  heads  E,  E  of  the  branches  D,  D.  These  are  free 
to  slide  in  the  sockets  F,  F,  so  that  measurements  may  be 
taken  between  points  not  accessible  by  instruments  with  fixed 
branches.  If  it  is  desired  to  find  the  width  of  an  opening,  the 
two  heads  E,  E  are  inserted  in  it,  and  then  separated  as  far  as 


8 


EXPERIMENTAL   PHYSICS. 


possible.     By  means  of  a  special  scale  and  vernier  on  the  arm 
AB,  readings  for  such  measurements  can  be  taken  directly. 


III.     THE   CATHETOMETER. 

The  cathetometer  is  used  to  measure  the  vertical  distance 
between  two  horizontal  planes  passing  through  two  points  not  in 
general  in  the  same  azimuth.  As  exhibited  in  Fig.  7,  it  consists 


Figf.  7- 


THE   CATHETOMETER.  9 

of  a  vertical  metal  cylinder  AB  with  a  scale  CD  inserted  in  it, 
is  movable  about  a  vertical  axis,  and  is  supported  on  a  platform 
furnished  with  three  adjustment  screws.  On  this  platform  are 
two  spirit  levels,  by  means  of  which  the  perpendicularity  of  AB 
may  be  tested.  Attached  to  this  vertical  column,  or  cylinder, 
is  a  carriage  EF,  upon  which  is  placed  a  telescope  with  its  opti- 
cal axis  horizontal.  This  carriage  is  capable  of  sliding  up  and 
down  the  column,  and  can  be  fixed  at  any  point  by  means  of  a 
compression  screw.  It  also  has  a  vernier  attached  to  it,  and 
carries  a  spirit  level  to  adjust  the  telescope. 

To  make  a  measurement,  sight  the  telescope  on  one  of  the 
points  so  that  its  image  coincides  with  the  intersection  of 
the  crosswires.  Note  the  reading  on  the  vernier,  and  then 
move  the  carriage,  and  the  telescope  up  or  down,  as  the  case 
requires,  until  the  image  of  the  other  point  coincides  with  the 
intersection  of  the  crosswires.  By  means  of  the  reading  then 
made,  and  the  previous  one,  the  vertical  distance  between  the 
points  may  be  found.  The  instrument  is  chiefly  used  in  finding 
the  heights  of  columns  of  mercury,  examples  of  which  are  found 
in  the  experiment  on  the  capillary  constant,  and  in  that  on  the 
absolute  expansion  of  mercury. 

If  the  two  points  are  at  a  considerable  distance  from  the 
cathetometer,  it  will  be  found  that  better  results  may  be 
obtained  by  placing  a  scale,  in  a  vertical  position,  alongside 
of  these  points,  and  then,  by  means  of  the  telescope  on  the 
instrument,  noting  what  division  on  the  scale  corresponds 
with  each  of  the  points  respectively.  In  the  construction  of 
the  instrument,  it  is  very  difficult  to  make  the  column  AB 
perfectly  rigid,  and  to  keep  it  so ;  and  from  this  cause  the 
readings  are  often  not  as  accurate  as  is  desirable.  The  instru- 
ment may  be  tested,  from  time  to  time,  by  measuring  the 
spaces  between  lines,  drawn  with  a  dividing  engine. 


I0  EXPERIMENTAL   PHYSICS, 

IV.     THE   SPHEROMETER. 

The  essential  part  of  the  spherometer  (Fig.  8)  is  a  microm- 
eter screw  DP,  which  passes  through  a  metal  tripod  whose  feet 
are  of  equal  length.  This  screw  carries  with  it,  at  its  upper 
end,  a  graduated  metallic  circular  disc 
LL',  whose  plane  is  perpendicular  to  the 
screw,  and  parallel  to  the  plane  passing 
through  the  three  terminals  A,  B,  C,  of 
the  supports.  To  one  of  these  supports 
there  is  attached  a  scale  RR'  graduated 
in  such  a  manner  that  on  causing  the 
disc  to  make  one  complete  revolution,  it 

Fig.  8. 

will  rise  or  fall  through  one  of  these  divi- 
sions. Usually  the  scale  is  divided  into  spaces  one-half  a  milli- 
meter in  width,  and  the  disc  into  500  equal  parts  so  that  readings 
can  be  taken  to  y^Vo  of  a  millimeter  when  any  particular  meas- 
urement is  about  to  be  made.  It  is  first  of  all  necessary  to  note 
the  reading  when  the  point  P  of  the  micrometer  screw  is  in 
the  same  plane  as  the  three  terminals  A,  B,  C,  and  the  instru- 
ment is  resting  equally  on  the  four  supports.  Usually  to 
the  beginner  this  presents  some  difficulty  ;  but  with  a  little 
practice,  and  by  repeatedly  grasping  one  of  the  supports  A, 
B,  and  C,  and  giving  the  instrument  a  slight  rotatory  move- 
ment, one  can  easily  perceive  when  contact  has  been  made  with 
the  screw.  If  the  instrument  is  standing  on  a  glass,  or  any 
reflecting  surface,  it  can  easily  be  ascertained  when  contact 
has  been  made  by  noting  the  instant  when  the  point  P  coincides 
with  its  image  as  seen  in  the  surface.  The  instrument  can  be 
used,  (i)  to  measure  small  thicknesses,  (2)  to  determine  the 
radius  of  curvature  of  a  sphere,  and  (3)  to  test  whether  a  surface 
is  perfectly  plane,  or  perfectly  spherical.  If  it  is  required  to 
measure  the  thickness  of  a  plate  with  parallel  faces,  first  place 
the  instrument  on  a  perfectly  plane  surface  (usually  made  of 
ground  glass),  and  turn  the  screw  until  its  point  P  comes  in 


THE   SPHEROMETER.  II 

contact  with  it ;  note  the  reading,  and  then  raise  the  screw  and 
place  under  the  point  P  the  plate  whose  thickness  is  to  be  deter- 
mined. Bring  the  point  of  the  screw  in  contact  with  the  upper 
surface  of  this  plate,  and  again  note  the  reading.  The  difference 
between  these  two  readings  will  give  its  thickness.  It  is  evident 
that  if  the  instrument  be  equally  supported  on  the  four  points 
A,  B,  C,  and  P  in  one  position,  and  then  caused  to  glide  over  a 
given  surface,  it  can  be  readily  seen  whether  the  surface  is  per- 
fectly plane,  or  not. 

To  find  the  radius  of  curvature  of  a  spherical  surface,  place 
the  spherometer  on  it,  and  ad- 
just the  micrometer  screw  until 
the  instrument  rests  equally  on 
the  four  supports.  Note  the 
reading,  and  then  place  the 
spherometer  on  a  plane  surface, 
and  again  establish  contact  at 
four  points,  and  note  the  read- 
ing. The  difference  between 
these  two  will  give  the  height  h 
of  the  spherical  segment  which 
has  for  its  base  a  plane  passing 
through  the  terminals  A,  B,  and  C,  when  the  first  reading  was 
taken. 

If  then  r  be  the  radius  of  the  base  of  this  segment,  h  its 
height,  and  a  the  radius  of  the  sphere,  we  have  from  Fig.  9, 


Fig.  9. 


2/1 


To  find  r,  place  the  instrument  on  a  sheet  of  cardboard  rest- 
ing on  the  three  supports,  and  the  screw,  and  press  it  lightly 
down  so  as  to  make  small  indentations.  Since  the  three  points 
A,  B,  and  C  form,  an  equilateral  triangle,  r  may  be  determined 


12 


EXPERIMENTAL   PHYSICS. 


by  taking  the  mean  of  the  distances  from  each  of  the  holes  made 
by  A,  B,  and  C  to  that  made  by  the  point  of  the  screw. 

Whether  a  surface  is  perfectly  spherical,  or  not,  may  be  de- 
termined by  adjusting  the  instrument  to  four-point  contact  in 
one  position,  and  then  sliding  it  over  the  surface. 

In  some  instruments  the  screw  is  hollow,  and  throughout  its 
length  a  slender  metallic  rod  runs.  This  rod  is  connected  to  a 
lever,  and  a  pointer,  which  indicates  at  once  the  instant  when 
contact  is  made. 


V.     MICROMETER    SCREW    GAUGE. 

This  gauge  (Fig.  10)  is  generally  used  to  measure  the  diam- 
eters of  fine  wires.  It  is  constructed  on  the  same  principle  as 
the  spherometer,  and  consists  of  a  bracket  or  shoulder  with  two 
arms  FE  and  FGt  in  the  former  of  which  is  screwed  a  small 
metal  cylinder  C,  with  its  protruding  face  perfectly  plane,  and 

7/7  /~V 

rj  Lr 


Fig.  10. 

at  right  angles  to  the  axis  of  the  micrometer  screw  AB,  which 
passes  through  a  threaded  tube  attached  to  the  arm  FG.  The 
end  of  the  screw  B  is  made  plane,  and  is  parallel  to  the  face  of 
the  cylinder  C.  To  the  other  end  there  is  attached  a  cap  A, 
which,  when  the  screw  is  turned,  comes  down  over  the  tube  on 
FG.  This  latter  has  a  scale  ruled  on  it,  and  the  divisions  are 
such  that  when  the  screw  is  made  to  turn  through  one  complete 
revolution,  the  cap  A  moves  over  one  of  the  divisions.  Just  as 


DIVIDING    ENGINE.  13 

in  the  case  of  the  disc  of  the  spherometer,  the  circumference  of 
the  cap  is  divided  into  a  number  of  equal  parts,  and  readings 
may  thus  be  made  to  the  fraction  of  a  division.  In  some  gauges 
the  scale  is  divided  into  millimeters,  and  the  circumference  of 
the  cap  into  twenty  equal  parts,  so  that  readings  may  be  taken 
to  2ao-  of  a  millimeter;  while  in  others  the  scale  is  divided  into 
spaces  of  ^  of  an  inch,  and  the  circumference  of  the  disc 
into  25  equal  parts.  In  this  latter  case,  readings  may  be  taken 
to  joVo  °f  an  mcn.  For  still  finer  readings  a  vernier  is  some- 
times ruled  longitudinally  on  the  back  of  the  tube,  in  the  man- 
ner shown  in  the  figure. 

In  making  a  measurement,  first  see  that,  when  the  screw  is 
in  contact  with  C,  the  zero  on  the  cap  corresponds  with  that  on 
the  scale  ;  if  not,  it  may  be  made  to  do  so  by  screwing  the  cyl- 
inder C  either  in  or  out,  as  the  case  requires.  The  wire  to  be 
measured  is  then  placed  between  the  face  of  C  and  that  of  the 
screw,  and  the  latter  turned  gently  until  it  is  felt  that  contact  is 
made.  The  reading  then  indicated  will  give  the  diameter  of 
the  wire.  Care  should  be  taken  to  insure  the  same  manner  of 
making  contact  while  measuring  the  wire  as  in  testing  for  the 
zero  reading:. 


VI.     DIVIDING    ENGINE. 

The  dividing  engine,  of  which  the  essential  parts  are  shewn  in 
Fig.  1 1,  is  employed  in  ruling  scales,  and  gratings,  and  in  measur- 
ing small  horizontal  distances.  It  embodies  the  same  principle 
as  the  spherometer,  and  consists  of  a  screw  B,  whose  pitch,  gen- 
erally one,  or  one-half  a  millimeter,  is  very  regular ;  a  divided 
circle  C  attached  to  one  end  of  it,  which  can  be  so  adjusted  that 
a  fractional  part  of  a  rotation  may  be  given  to  the  screw  ;  and  a 
carriage  EE,  which  slides  along  the  smooth  rail  /  attached  to 
the  solid  base  A.  This  carriage  EE,  on  which  are  supported  a 
tracing  point  .Fwith  a  handle  G,  and  a  microscope  H,  by  means 
of  which  the  rulings  are  examined,  may  be  thrown  in  or  out  of 


EXPERIMENTAL   PHYSICS. 


gear  with  the  screw  B  by  turning  the  rod  K,  which  opens  the 
clutch  shown  in  the  figure. 

The  object  which  is  to  be  ruled,  or  measured,  is  placed  on 


SPECIFIC    GRAVITY    BOTTLE.  15 

the  platform  J  directly  beneath  the  tracing  point,  and  the  micro- 
scope. 

Scales  on  metal  may  be  ruled  either  directly  by  using  a 
hardened  steel  tracing  point,  or  by  ruling  on  a  wax-covered 
surface,  and  then  etching  in  with  acid.  This  latter  method  may 
also  be  adopted  in  the  case  of  glass  scales,  but  these  are  gener- 
ally made  by  using  a  diamond  tracing  point.  For  such  fine 
work  as  ruling  gratings,  dividing  engines  are  specially'  con- 
structed with  screws  whose  pitch  depends  on  the  degree  of 
delicacy  required  in  the  work. 

VII.     SPECIFIC   GRAVITY   BOTTLE. 

Figure  12  shows  the  specific  gravity  bottle,  with  ground  glass 
stopper,  generally  used  in  laboratory  experiments.  Throughout 
the  length  of  the  stopper,  a  very  fine  hole  runs,  which  is  ter- 
minated in  a  small  cup-shaped  opening  at  the  top. 
A  mark  is  made  on  the  narrow  part  of  the  stopper, 
and  enough  liquid  is  put  in  the  bottle  to  make  it 
rise  to  this  point  when  the  stopper  is  in  tight.  In 
what  follows,  the  bottle  will  always  be  considered 
full  when  the  surface  of  the  liquid  is  in  this 
position. 

Great  care  should  be  taken  to  have  the  bottle 
perfectly  clean.  This  is  done  by  first  washing  it 
out  with  a  solution  of  caustic  potash,  and  then 
with  hydrochloric  acid.  After  this,  wash  it  several  times  with 
ordinary  water,  and  then  rinse  out  with  distilled  water,  and  place 
it  in  a  hot  air  bath  to  dry.  In  filling  the  bottle  with  a  liquid, 
the  student  should  be  careful  to  have  all  the  air  bubbles 
removed. 

EXERCISE    I.  —  To  find  the  specific  gravity  of  a  liquid. 

Let  w  be  the  weight  of  the  bottle  in  air,  w'  its  weight  when 
filled  with  distilled  water,  and  w"  when  filled  with  the  given 
liquid.  Equal  volumes  of  the  liquid,  and  of  the  water  weigh 


l6  EXPERIMENTAL   PHYSICS. 

then  w" '  —  w,  and  wr  —  w,  respectively,  and  the  specific  gravity 
of  the  former  is  therefore  given  by  w  f  ~w. 


EXERCISE  II.  —  To  find  the  specific  gravity  of  a  solid  broken 
up  into  small  pieces. 

Let  w  be  the  weight  of  the  pieces  of  the  solid,  w!  the  weight 
of  the  bottle  rilled  with  distilled  water,  and  w"  the  weight  of  the 
bottle  partly  filled  with  the  pieces,  and  the  balance  filled  with 
water.  Then  w  +  w'  —  w"  will  be  the  weight  of  the  water  dis- 
placed by  the  pieces,  and  the  specific  gravity  of  the  solid  is 
therefore 


w  +  w'  —  w" 

If  the  solid  is  soluble  in  water,  the  same  method  may  be  used 
to  find  its  specific  gravity  relative  to  some  liquid  in  which  it  is 
not  soluble,  and  then,  the  specific  gravity  of  this  liquid  having 
been  determined,  that  of  the  solid  may  be  calculated. 

A  suitable  exercise  on  this  method  is  to  find  the  specific 
gravity  of  wires  used  in  the  experiments  on  the  sonometer.  In 
this  case  the  wire  should  be  cut  up  into  small  pieces  unless  it  is 
very  fine. 

If  very  great  accuracy  is  required  in  these  experiments,  cor- 
rections will  have  to  be  made  for  temperature,  and  for  the 
buoyancy  of  the  air. 

VIII.     HYDROSTATIC   BALANCE. 

Experiment  I.  —  In  Fig.  13  a  balance  is  shewn  whose  scale 
pans  are  at  a  considerable  height  above  the  platform  on  which 
it  stands.  To  one  of  these  pans,  as  shewn  in  the  figure,  attach 
a  cylindrical  cup  A,  and  also  the  cylindrical  solid  B,  which  just 
fills  this  cup.  Then  balance  the  two  by  weights  placed  in  the 
opposite  pan.  Having  done  this,  place  a  beaker  of  water  under 
A  and  B,  so  that  B  is  completely  submerged,  and  shew  that  on 
filling  A  with  water  equilibrium  is  restored.  This  experiment 


HYDROSTATIC    BALANCE.  17 

shews  that  the  resultant  vertical  pressure  on  the  cylinder  B  is 
equal  to  the  weight  of  the  water  it  displaces,  and  this  is  the 
principle  upon  which  the  following  methods  are  based. 

EXERCISE  I.  —  To  find  the  specific  gravity  of  a  solid  insoluble 
in  water,  and  heavier  than  water. 

To  one  of  the  arms  of  the  balance  suspend  the  given  solid  by 
a  very  fine  thread.     Let  w  be  its  weight  in  air,  and  w'  its  weight 


Fig.  13. 


when  submerged  in  water.     The  weight  of  the  water  displaced 

by  the  solid  will  then  be  w  —  w',  and  therefore  — ^—.  is  its 

w  —  w 

specific  gravity. 

EXERCISE  II.  —  To  find  the  specific  gravity  of  a  liquid. 

Suspend,  as  in  Exercise  I.,  from  one  of  the  scale  pans  a  solid 
heavier  than  water,  or  the  given  liquid,  and  insoluble  in  both. 
Let  the  weight  of  the  solid  when  suspended  in  air  be  w,  in 
water  w',  and  in  the  given  liquid  w". 

The  weight  of  the  water  displaced  by  the  solid  will  therefore 


18 


EXPERIMENTAL   PHYSICS. 


be  w  —  iv',  and  that  of  the  liquid  displaced  by  it  w  —  w" .     The 


w  —  w 
specific  gravity  of  the  liquid  is  then  — — — , 


it 

w' 


EXERCISE  III.  —  To  find  the  specific  gravity  of  a  solid  lighter 
than  water,  and  insoluble  in  it. 

In  this  exercise  it  is  necessary  to  use  some  such  substance  as 
a  piece  of  iron  in  order  to  cause  the  given  solid  to  sink  in  the 
water. 

Let  w  and  w'  be  the  weights  of  the  given  solid,  and  the  sinker 
respectively  in  air,  w"  that  of  both  together  in  water,  and  w'" 
that  of  the  sinker  alone  in  water. 

Then  w'  —  w"'  is  the  weight  of  the  water  displaced  by  the 
sinker,  and  w  +  w'  —  w "  that  of  the  water  displaced  by  both 
the  solid  and  the  sinker.  The  weight  of  the  water  displaced  by 
the  solid  is  therefore  w  —  w"  +  w'"t  and  its  specific  gravity  is 

then  given  by  —     — r. 777- 

w  —  w    -f-  w 

Experiment  II.  —  Place  a  beaker  of  water  on  one  of  the 
scale  pans  of  an  ordinary  balance  as  in  Fig.  14,  and  on  the 


Fig.  14. 

other  place  the  cup  A,  and  sufficient  weights  to  produce  equi- 
librium. Then  suspend  in  the  water,  in  the  manner  indicated, 
the  cylindrical  solid  B,  and  shew  that  on  filling  the  cup  A  with 


NICHOLSON'S   HYDROMETER.  19 

water  equilibrium  is  restored.  This  experiment  shews  that 
the  reaction  of  the  cylinder  B  on  the  water,  which  is  equal  to 
the  weight  of  the  water  it  displaces,  is  equal  to  the  action  of  the 
water  on  the  cylinder.  By  adopting  this  method,  the  weight  of 
the  water  displaced  by  a  given  solid  may  be  found  directly,  and 
it  will  be  very  instructive  for  the  student  to  use  it  in  finding  the 
specific  gravities  indicated  in  Exercises  I.,  II.,  and  III. 


IX.  NICHOLSON'S  HYDROMETER. 

The  apparatus  (Fig.  15)  consists  of  a  hollow  metallic  vessel 
B,  to  one  end  of  which  is  attached  a  slender  stem  terminated  in 
a  pan  D,  capable  of  holding  weights,  or  the  solid  whose  specific 
gravity  is  to  be  determined.  To  the 
other  end  of  the  instrument  there  is 
attached  a  second  platform,  or  pan  C, 
made  very  heavy,  so  that  when  the 
instrument  is  at  rest  in  water  it  will 
assume  the  position  indicated  in  the 
figure. 

When  placed  in  water,  the  instru- 
ment sinks  until  it  is  only  partially 
submerged,  and  by  placing  weights  on 
the  pan  D,  it  may  be  made  to  sink  to 
any  required  depth. 

To  find  the  specific  gravity  of  a  solid 
insoluble  in  water. 

Let  the  weight  />,  when  put  on  the 
pan  D,  sink  the  instrument  to  the  mark 
A  on  the  stem,  in  pure  distilled  water. 

Remove  this  weight,  and  place  the  given  solid  on  the  pan. 
Let  /  be  the  weight  that  must  now  be  added  to  cause  the 
hydrometer  to  sink  to  the  mark  A  ;  P  —p  will  therefore  be 
the  weight  of  the  solid.  Leave  the  weight  p  on  the  pan  D, 


20  EXPERIMENTAL   PHYSICS. 

and  place  the  solid  then  on  the  pan  C,  and  since  the  solid 
will  now  lose  a  part  of  its  weight  equal  to  that  of  the  water 
it  displaces,  more  weights  must  be  placed  on  the  pan  D  to  sink 
the  instrument  to  the  mark  A.  Let  P'  be  the  amount  of  these 
weights.  Then  P'  will  be  the  weight  of  the  water  displaced  by 
the  solid ;  and,  since  its  own  weight  is  P  — /,  its  specific  gravity 

is  therefore  ~ 

If  the  solid  used  is  lighter  than  water,  it  will  tend  to  rise 
when  placed  on  the  pan  C.  In  this  case  it  is  prevented  from 
rising  by  being  placed  in  a  wire  cage  attached  to  C,  and  the 
experiment  is  conducted  exactly  in  the  manner  indicated  above. 
If  the  solid  to  be  tested  is  soluble  in  water,  the  experiment  may 
be  conducted  by  using  some  liquid  of  known  specific  gravity  in 
which  it  is  not  soluble. 

If  the  stem  of  the  instrument  be  made  very  slender,  the 
hydrometer  will  be  sensitive  to  small  variations  in  the  weights 
placed  on  the  pan  ;  but  owing  to  the  surface  tension  of  the 
water  on  this  stem,  and  on  the  walls  of  the  instrument,  the 
results  obtained  will  not  be  as  accurate  as  those  obtained  when 
either  of  the  two  previous  methods  is  adopted.  The  instru- 
ment, however,  enables  us  to  find  rapidly  the  specific  gravity  of 
a  solid  approximately.  Corrections  may  also  be  made  in  this 
experiment  for  the  temperature  of  the  water. 

X.  FAHRENHEIT'S  HYDROMETER. 

Fahrenheit's  hydrometer  (Fig.  16)  embodies  the  same  prin- 
ciple as  that  of  Nicholson,  and  resembles  it  in  form,  except  that 
it  is  made  of  glass  so  that  it  may  be  used  in  all  liquids.  To  its 
lower  extremity  there  is  attached  a  small  glass  bulb  generally 
loaded  with  mercury  or  small  shot.  This  causes  the  instrument 
to  float  in  an  erect  position. 

To  find  the  specific  gravity  of  a  liquid. 

Let  P  be  the  weight  of  the  instrument  when  weighed  in  air, 
and  p  the  weight  which,  when  placed  on  the  pan,  causes  it  to 


FAHRENHEIT'S  HYDROMETER. 


21 


sink  in  pure  distilled  water  to  the  mark  A  on  the  stem.  Let  /' 
be  the  weight  which  must  be  placed  in  the  pan  to  cause  it  to 
sink  to  this  mark  when  placed  in  the  given  liquid.  Then,  since 
P+P,  and  P+  f'  are  the  weights  of  equal  volumes  of  water,  and 
of  the  given  liquid  respectively,  the  specific  gravity  of  the  latter 

is  therefore  —    **  • 


Hydrometers  of  constant  immersion,  such  as  those  of  Nichol- 
son, and  Fahrenheit,  are  but  little  used  outside  of  laboratories.  In 
actual  practice  instruments  similar  to  those  exhibited  in  Fig.  17 


Fig.  16. 


Fig.  17. 


are  used.  They  sink  to  different  levels  when  placed  in  differ- 
ent liquids,  and  from  the  graduations  of  the  scales  attached  to 
their  stems  the  specific  gravity  of  the  liquid  tested  may  be 
determined.  Instead  of  indicating  specific  gravities,  these  scales 
are  sometimes  graduated  to  indicate  percentages  of  some  sub- 
stance mixed  with,  or  dissolved  in  a  liquid,  such  as  alcohol, 
sugar,  or  salt,  in  water,  or  butter  fat  in  milk.  When  so 
used  the  instruments  are  called  alcoholometers,  densimeters, 
salimeters,  or  lactometers,  according  to  the  use  to  which  they 
are  to  be  put. 


22 


EXPERIMENTAL   PHYSICS. 


They  are  sometimes  still  further  distinguished  by  means  of 
the  name  of  the  manufacturer,  those  generally  used  being 
Beaume's,  and  Tweedel's.  Small,  hollow  glass  balls  are  some- 
times used  to  find  the  specific  gravity  of  a  liquid.  They  are 
made  in  a  graded  series,  and  numbers  are  cut  in  the  glass, 
giving  their  specific  gravities,  which  have  been  determined  by 
using  them  in  liquids  whose  specific  gravities  are  known.  To 
find  the  specific  gravity  of  a  liquid,  it  is  only  necessary  to  find 
which  one  of  the  balls  will  just  float  in  it. 

By  experimenting  with  different  solutions,  and  mixtures,  the 
student  should  make  himself  thoroughly  familiar  with  the  use 
of  these  practical  instruments. 


XI.     MOHR'S   BALANCE. 


Many  instruments  have  been  constructed  for  the  rapid  deter- 
mination of  the  specific  gravity  of  liquids,  and  of  these  the 
Westphal  modification  of  Mohr's  balance  shewn  in  Fig.  18  is 


Fig.  18. 

probably  the  most  convenient.  It  consists  of  a  weighing  beam 
AB  with  knife  edges  resting  on  the  stand  C,  a  glass  thermom- 
eter plummet  D  suspended  from  it  by  a  platinum  wire,  and  a  set 


MOHR'S    BALANCE.  23 

of  four  riders  E,  E,  E,  E,  whose  weights  are  5,  .5,  .05,  and  .005 
grams  respectively.  The  beam  AB  is  graduated  in  ten  equal 
divisions,  and  the  glass  plummet,  whose  weight  is  15  grams,  is 
of  such  a  volume  that  it  displaces  5  grams  of  distilled  water  at 
16.7°  C. 

If,  then,  equilibrium  exists  with  the  plummet  attached  to  the 
beam  in  air,  it  can  be  readily  seen  that  it  will  again  be  estab- 
lished, if  the  plummet  be  immersed  in  distilled  water  at  16.7°  C, 
by  hanging  the  largest,  or  unity  rider  to  the  hook  on  the  end  of 
the  beam.  If  the  plummet  be  immersed  in  a  liquid  lighter  than 
distilled  water,  the  unity  rider  is  removed  from  the  hook,  and 
placed  upon  that  point  on  the  beam  which  will  bring  it  into 
equilibrium.  If  this  point  lies  between  two  division  marks,  the 
unity  rider  is  placed  upon  the  lower  of  these,  and  the  point  of 
equilibrium  is  sought  by  a  similar  application  of  the  next  smaller 
rider,  and  also,  if  necessary,  of  the  two  following  ones.  If  it 
should  happen  that  two  riders  come  to  find  their  places  on  the 
same  division  mark,  the  smaller  weight  is  hung  upon  the  larger. 
If  when  the  plummet  is  immersed  in  such  a  liquid  the  riders 
in  order  of  size  are  on  the  points  9,  2,  7,  and  6  respectively, 
the  student  can  easily  deduce  from  elementary  principles  of 
mechanics  that  the  weight  of  the  liquid  displaced  by  the  plum- 
met is  .9276  that  of  the  distilled  water  displaced  by  it ;  the 
specific  gravity  of  the  liquid  examined  is  then  .9276. 

In  determining  the  specific  gravity  of  a  liquid  heaver  than 
water,  the  unity  rider  is  allowed  to  remain  on  the  hook  with  the 
plummet,  while  a  second  unity  rider,  and  the  three  other  smaller 
riders  are  used  on  the  beam  as  described  above. 

As  the  instrument  illustrates  splendidly  the  principles  of 
mechanics  and  hydrostatics,  the  student  should  make  himself 
thoroughly  familiar  with  its  workings. 


EXPERIMENTAL   PHYSICS. 


XII.     BOYLE'S   LAW. 
A.    For  pressures  greater  than  one  atmosphere. 

In  Fig.  19,  CD  and  BF  are  two  glass  tubes  attached  to  a 
stand,  and  connected  at  the  ends  D,  and  B  by  a  metal  tube  into 
which  both  are  cemented.  Between  them  a  scale  is  so  placed 
as  to  indicate  equal  volumes  in  the  tube  BF.  The  upper  end 
of  this  tube  is  closed  by  two  taps  F,  and  G;  but  if  the  ex- 
periment is  conducted  with  a  dry  gas,  only  one  of  these  is  used, 
the  upper  one,  G,  being  then  removed. 


Experiment.  —  Open  the  tap  F  so  as  to 
allow  the  air  in  the  tube  BF  to  communi- 
cate with  that  outside,  and  pour  mercury 
into  the  open  end  C  of  the  tube  CD  until 
it  rises  in  both  tubes  to  the  level  of  some 
chosen  mark  on  the  scale.  By  then  closing 
the  tap  F  a  mass  of  air  will  be  inclosed 
in  the  tube  BF  at  atmospheric  pressure. 
On  pouring  more  mercury  in  at  C,  the 
volume  occupied  by  the  air  will  be  gradu- 
ally decreased,  and  the  mercury  will  rise 
to  different  heights  in  the  two  tubes.  Let 
//be  the  height  of  the  barometric  column, 
and  Fthe  volume  occupied  by  the  air  at  at- 
H|  mospheric  pressure.  When  h  is  the  differ- 
ence between  the  heights  of  the  two  col- 
umns of  mercury,  let  v  be  the  volume  of 
the  air/  The  pressures,  therefore,  to  which 


Fig.  19- 


the  air  is  subjected  in  the  two  cases  are  proportional  to  //,  and 
H+h  respectively.  If  the  experiment  is  conducted  carefully,  it 
will  be  found  that  HV=(H+h)v,  which  proves  the  law  that 
the  volume  of  a  mass  of  dry  air  at  constant  temperature  varies 
inversely  as  the  pressure  to  which  it  is  subjected.  The  tube 


BOYLE'S    LAW. 


BF  may  be  filled  with  dry  air  before  commencing  the  experi- 
ment, by  first  filling  the  two  tubes  with  mercury,  and  then 
letting  it  run  out  through  the  tap  at  the  bottom  of  the  metal 
tube,  the  air  entering  BF  at  the  tap  F  after  passing  through  a 
drying  tube  filled  with  calcium  chloride. 

The  same  experiment  may  be  conducted  with  other  gases, 
and  it  will  be  found  that  they  follow  the  same  law.  By  attach- 
ing the  tap  G  (shown  in  section  at  O,  and  O')  to  the  tube  BF, 
we  may,  on  account  of  its  peculiar  construc- 
tion, admit  a  liquid  to  the  tube  drop  by 
drop,  and  the  laws  of  superheated,  and 
saturated  vapours  may  thus  be  investigated. 

B.    For  pressures  less  than  one  atmosphere. 
In  Fig.  20,  PM,  a  long,  narrow  vessel,  sup- 
ported on  a  stand,  is  nearly  filled  with  mer- 
cury, and  AB,  a  glass  tube  closed  at  one  end, 
is  graduated  to  indicate  equal  volumes. 

Experiment.  —  Partly  fill  the  tube  AB  with 
mercury,  invert  it  in  the  vessel  PM,  and 
then  lower  it  until  the  surfaces  of  the  mer- 
cury inside,  and  outside  the  tube,  are  in  the 
same  plane.  The  air  in  the  tube  is  then 
at  atmospheric  pressure.  Let  V  be  the 
volume  it  then  occupies,  and  H  the  height 
of  the  barometric  column.  Raise  the  tube 
little  by  little,  and  note  each  time  the 
volume  the  air  occupies  and  the  pressure 
to  which  it  is  subjected.  In  the  figure 
AC  is  the  volume  of  the  air,  and  CD  the  height  of  the  mer- 
cury in  the  tube  at  one  stage  of  the  experiment.  Since  the 
pressure  is  the  same  inside,  and  outside  the  tube  at  the  level  D, 
it  is  readily  seen  that  the  pressure  to  which  the  air  is  subjected 
in  this  case  is  proportional  to  the  difference  between  the  height 


26 


EXPERIMENTAL    PHYSICS. 


CD,  and  that  of  the  barometer.  If  CD  is  denoted  by  7i,  this 
difference  is  then  equal  to  H—h.  Here  again,  denoting  the 
volume  AC  by  v,  we  will  have  H V  equal  to  (H—k}v,  and  will 
thus  find  that  the  law  holds  for 
pressures  less  than  one  atmos- 
phere. 

Usually  there  is  also  a  baro- 
metric tube  inverted  in  the  mer- 
cury in  PM,  and  attached  to  a 
support.  The  quantity  (H '—  Ji) 
can  then  be  measured  directly,  and 
accurately  by  means  of  a  cathe- 
tometer. 


XIII.     THE   VOLUMENOMETER. 

This  instrument  is  used  to  find 
the  volume  of  a  substance,  such 
as  gunpowder,  sugar,  or  salt,  which 
cannot  be  placed  in  contact  with 
water.  In  Fig.  21,  AF,  and  BG 
are  two  glass  tubes  containing 
mercury,  whose  ends  F  and  G  are 
cemented  into  a  metal  tube  which 
connects  them.  In  this  tube  there 
is  a  three-way  tap  which  serves 
different  purposes,  according  to 
the  position  in  which  it  is  placed. 
These  are  indicated  in  Figs,  (i), 
(2),  (3),  and  (4). 
In  (i)  communication  is  established  between  the  two  tubes, 

in  (2)  the  mercury  can  run  out  of  both  tubes,  in  (3)  it  can  run 

out  of  ^.Fonly,  and  in  (4)  out  of  BG  only. 

On  the  tube  BG  are  two  marks,  B  and  K,  and  between  them 

the  tube  expands  into  a  little  bulb ;  above  B  the  tube  narrows 


Fig.  21. 


THE   VOLUMENOMETER.  27 

down,  and  after  being  bent  is  connected  to  a  three-way  socket,  at 
one  end  of  which  is  a  tap  E,  and  to  the  other  there  is  attached 
a  glass  globe  which  can  be  readily  removed,  or  put  on  by  means 
of  the  collar  D. 

Let  v  be  the  volume  of  the  tube  between  the  marks  B  and  K, 
Fthat  of  the  tube  above  B  including  the  globe  attached  at  D, 
and  x  the  required  volume  of  the  given  substance.  To  calculate 
v,  fill  the  tube  BG  with  mercury  up  to  the  mark  B,  and  then 
let  it  run  out  through  the  tap  H  until  it  sinks  in  the  tube  to  the 
mark  K.  By  weighing  the  mercury  that  runs  out  v  may  be 
determined. 

Experiment.  —  Attach  the  globe  at  D,  and  after  opening  the 
tap  E  fill  the  two  tubes  with  mercury  up  to  the  mark  B.  Close 
the  tap  E,  the  air  thus  shut  in  being  at  atmospheric  pressure, 
and  then  allow  the  mercury  to  run  out  of  both  tubes  until  it  is 
at  the  mark  K  in  the  tube  BG.  Let  h  be  the  height  of  the 
mercury  in  this  tube  above  that  in  the  tube  AF,  and  applying 
Boyle's  Law,  we  have  therefore,  if  H  is  the  barometric  height, 

HV=(H-h)(V+v).  (i) 

Now  remove  the  globe,  put  the  given  substance  whose  volume 
is  to  be  determined  into  it,  and  after  again  attaching  it  by  the 
collar  D,  repeat  the  operation  exactly  as  described  above.  If  h' 
be  now  the  height  of  the  mercury  in  one  tube  above  that  in  the 
other,  and  H'  that  of  the  barometric  column,  we  have  again, 
by  applying  Boyle's  Law, 


x  +  v}.  (2) 

From  (i)  we  have        y  =          -  (3) 


(4) 


28  EXPERIMENTAL   PHYSICS. 

.'.  from  (3)  and  (4) 


x=v\-— 

Ji     k 


vH(h'- 


If  Wis  the  weight  of  the  substance  in  grams,  and  x  its  vol- 
ume in  cubic  centimeters,  its  specific  gravity  is  given  by 

c      r         W 
Sp.  Gr.  =  — 

X 

The  two  tubes  AF,  and  BG  are  usually  graduated  in  centi- 
meters, and  in  this  case  h  and  //  can  be  read  off  directly. 
Otherwise  a  cathetometer  must  be  used. 


XIV.     DETERMINATION    OF   CAPILLARY   CONSTANTS. 

Various  phenomena  in  nature  show  that  the  molecules  of  a 
body,  whether  in  a  solid  or  in  a  liquid  state,  are  subject  to  the 
action  of  two  contrary  forces,  one  of  which  tends  to  bring  them 
together,  and  the  other  to  separate  them  from  each  other.  The 


Fig.  22. 


former  of  these  is  a  cohesive  or  attractive  force  (quite  distinct 
from  gravitation),  which  acts  only  at  very  short  distances  be- 
tween particles  of  the  same,  or  of  different  matter ;  while  the 


DETERMINATION   OF   CAPILLARY   CONSTANTS.  29 

latter  is  due  to  a  vibratory  motion  which  the  molecules  are  sup- 
posed to  receive  on  the  application  of  heat.  It  is  with  these 
attractive  forces,  whose  presence  is  the  condition  for  the  exist- 
ence of  matter  in  the  solid  state,  that  we  wish  especially  to  deal. 

That  they  act  only  at  very  short  distances  is  evidenced  by  the 
fact,  that,  if  we  grind  up  a  solid  into  a  fine  powder,  the  particles, 
when  the  matter  is  in  this  state,  are  so  far  apart  that  these 
forces  exert  no  influence  ;  while  if  we  take  two  pieces  of  lead 
or  glass  with  very  smooth  plane  surfaces,  and  place  these  in 
contact,  we  find  that  they  at  once  adhere  to  each  other,  and  it 
is  only  on  the  application  of  considerable  force  that  we  are  able 
to  separate  them.  Although  the  presence  of  these  forces  is 
chiefly  manifested  in  matter  in  the  solid  state,  they  exert,  how- 
ever, an  action  between  the  particles  of  a  liquid,  and  the  general 
term  capillarity  is  used  to  denote  all  the  phenomena  produced 
by  them. 

If  we  dip  a  clean  glass  rod  into  water,  we  find  on  withdrawing 
it  that  a  drop  clings  to  its  lower  extremity.  As  gravity  tends 
to  detach  the  particles  of  the  drop  from  each  other,  and  these 
again  from  the  glass,  we  learn  that  there  is  an  attraction  between 
the  particles  of  water  forming  the  drop,  and  also  one  between 
the  particles  of  glass,  and  of  water,  which  is  greater  than  that 
exerted  by  gravity.  Again,  if  we  suspend  a  glass  disc  horizon- 
tally from  one  of  the  arms  of  a  balance,  and  allow  it  to  come  in 
contact  with  the  surface  of  water,  we  find  that  in  order  to  detach 
it  a  considerable  number  of  weights  must  be  placed  in  the  pan 
attached  to  the  opposite  arm.  If  instead  of  water  we  use 
mercury,  exactly  the  same  action  takes  place,  but  a  smaller 
weight  is  necessary  to  make  the  separation.  If  we  replace  the 
glass  disc  by  one  of  copper  of  exactly  the  same  form,  and  dimen- 
sions, we  will  find  that  the  same  weight  as  before  must  be  used 
to  detach  it  from  the  water,  but  one  different  from  that  used 
before  to  make  the  separation  from  mercury.  The  explanation 
is  simple.  In  the  one  case  a  layer  of  water  adheres  to  the  disc, 
and  we  separate,  not  the  disc  from  the  water,  but  a  layer  of  water 


30  EXPERIMENTAL   PHYSICS. 

from  the  main  body ;  while  with  mercury  we  separate  the  disc 
from  the  liquid.  This  shews  that  the  attractions  between  par- 
ticles of  water  are  less  than  those  between  particles  of  glass, 
and  of  water,  and  it  affords  an  explanation  of  the  term  wetting. 
A  liquid  is  said  to  wet  a  solid  when  the  particles  of  the  solid 
attract  those  of  the  liquid  with  a  force  greater  than  that  exerted 
between  the  particles  of  liquid  themselves. 

Another  class  of  phenomena  caused  by  these  molecular  actions 
is  exhibited  in  the  following  experiments.  If  we  make  a  small 
cup  out  of  a  sheet  of  wire  gauze,  and,  after  dipping  it  in  melted 
paraffine,  shake  it  so  that  its  meshes  do  not  become  stopped  up, 
we  will  find  that  we  can  pour  into  it  a  large  quantity  of  water 
before  any  begins  to  run  out.  Or,  again,  if  we  rub  a  common 
sewing  needle  with  grease  so  that  the  water  will  not  wet  it,  and 
then  place  it  gently  on  the  surface  of  clean  water,  it  will  float 
there  if  not  disturbed.  These,  and  many  other  such  experi- 
ments, lead  us  to  conclude  that  at  the  surface  of  a  liquid  there 
is  a  thin  layer  of  particles  which  act  in  the  same  manner  as  a 
flexible  elastic  membrane  spread  over  the  surface.  Although 
it  is  difficult  to  explain  just  why  this  outside  layer  acts  in  this 
manner,  we  can,  however,  readily  understand  that  the  particles 
near  the  surface  of  a  liquid  are  subject  to  actions  different  in 
degree  from  those  on  the  particles  in  the  interior. 

In  Fig.  23  let  ABC  represent  a  mass  of  liquid,  and  P  a  par- 
ticle of  it  situated  at  the  center  of  a  small  sphere  DE  of  such  a 
radius  that  none  of  the  liquid  outside  of  it  has  any  action  on 
the  particle  at  P.  If  then  P  is  at  a  point  in  the  liquid  whose 
distance  from  the  surface  is  greater  than  the  radius  of  this 
sphere,  it  is  evident,  from  symmetry,  that  the  resultant  action  on 
the  particle  is  zero.  If,  however,  it  is  situated  at  Pv  it  is  clear 
that  there  is  no  liquid  to  compensate  the  action  of  the  part  FGH 
of  the  sphere,  and  there  will  be,  in  this  case,  a  resultant 
action  on  P1  depending  on  the  mass  FGH,  and  tending  to  draw 
it  into  the  body  of  the  liquid.  This  will  be  still  more  evi- 
dent if  the  particle  is  situated  in  the  surface,  at  P2;  for  in 


DETERMINATION   OF   CAPILLARY   CONSTANTS.  31 

this  case  there  is  the  action  of  a  hemisphere  of  liquid  not 
compensated. 

We  thus  see  that  the  particles  at,  or  near,  the  surface  of  a 
liquid  are  subjected  to  attractions  which  tend  to  draw  them  into 
the  main  body,  and  the  effect  on  the  whole  mass  is  precisely 
the  same  as  if  it  were  surrounded  by  a  thin  elastic  membrane. 

In  the  case  of  liquid  films,  and  soap  bubbles  we  have  two 
sets  of  surface  particles  with  a  thin  layer  of  liquid  between  them. 
Delicate  experiments  have  been  devised  with  these  to  ascertain 
the  force  required  to  separate  the  particles  in  the  surface  layer 


Fig.  23. 

from  each  other  by  measuring  the  radii  of  bubbles  just  before 
they  break,  and  by  noting  the  pressures  to  which  they  are  sub- 
jected ;  but  probably  it  can  best  be  found  in  the  case  of  liquids 
which  wet  glass  by  the  use  of  capillary  tubes.  Figure  22 
shows  a  glass  vessel  nearly  filled  with  such  a  liquid  in  which 
are  suspended  several  capillary  tubes  A,  B,  C,  and  D,  a  sharp- 
pointed  screw  E  which  can  be  raised  or  lowered,  and  a  cathe- 
tometer  L,  If  the  liquid  be  drawn  up  through  one  of  these 
tubes,  and  then  allowed  to  fall  back,  it  will  leave  a  layer  of  its 
particles  adhering  to  the  glass  on  the  inside,  and  will  not  sink 
to  the  level  of  the  outside  liquid,  but  will  remain  stationary  at 
a  height  depending  on  the  radius  of  the  tube. 

Since  the  hydrostatic  pressure  inside  the  tube  is  the  same  as 
that  outside  for  points  in  the  same  horizontal  plane,  then  the 
pressure  inside  just  below  the  surface  layer  of  particles  is  less 


32  EXPERIMENTAL  PHYSICS. 

than  that  at  the  surface  of  the  main  body  of  the  liquid,  and  we 
may  therefore  look  upon  the  column  as  being  held  up  by  the 
attractions  between  the  particles  along  the  periphery  of  the  sur- 
face layer,  and  those  adhering  to  the  tube.  If,  then,  T  denotes 
the  value  of  this  surface  tension  per  unit  length,  h  the  height 
of  the  column,  and  r  its  radius,  we  have,  since  the  angle  at 
which  the  surface  of  the  liquid  meets  the  tube  is  zero, 


or  T=gP.rh. 

As  this  tension  T  is  exerted  between  particles  of  the  same 
kind  of  matter,  it  should  have  a  constant  value  whatever  may  be 
the  radius  of  the  tube.  The  product,  rh,  therefore  should  be  a 
constant,  and  this  may  be  illustrated  by  using  tubes  of  different 
sizes. 

In  order  to  ascertain  the  height  h  at  which  the  liquid  stands, 
lower  the  screw  E  until  it  just  grazes  the  liquid,  and  then  with 
the  cathetometer  find  the  vertical  distance  between  the  surface 
of  the  liquid  in  the  tube,  and  the  upper  extremity  of  the  screw. 
The  difference  between  this  length,  and  that  of  the  screw,  which 
has  been  previously  determined,  will  give  h. 

The  radius  of  the  tube  r  is  found  by  running  a  thread  of 
mercury  into  it.  If  /  be  the  length  of  this  thread  in  centimeters, 
w  its  weight  in  grams,  and  d  the  density  of  mercury,  then 


=  w, 


or 


irld 


This  method  of  finding  the  surface  tension  may  be  adopted 
in  the  case  of  such  liquids  as  water,  alcohol,  or  sulphuric  acid, 
and  in  order  to  obtain  uniform  results,  the  greatest  care  must  be 
taken  to  have  tubes,  and  liquid  perfectly  clean. 

There  are  many  instructive  experiments  in  capillarity,  and 
students  should  especially  make  themselves  familiar  with  those 


THE    SEXTANT.  33 

devised   by    Plateau   on  liquid  films,  and  on  the  forms  that  a 
mass  of  liquid  will  take  when  freed  from  the  action  of  gravity. 

A  full  account  of  the  various  elementary  phenomena  of  capil- 
larity is  to  be  found  in  a  little  "book  entitled  "  Soap  Bubbles  " 
by  C.  V.  Boys. 

XV.     THE    SEXTANT. 

This  instrument,  shewn  in  Fig.  24,  consists  of  a  framework 
in  the  shape  of  a  circular  sector,  of  which  the  arc  GH  is  gradu- 
ated. About  the  center  of  this  sector  turns  an  index  arm  EF 
provided  with  a  vernier  and  tangent  screw.  At  the  center 
a  mirror  M  is  fixed  normally  to  this  arm,  and  moves  with 


M 


F 


Fig.  24. 


it ;  while  to  the  frame  on  the  left-hand  side  is  attached  a 
second  mirror  M',  which  has  only  the  lower  half  silvered,  and 
may  by  means  of  screws  be  rotated  about  an  axis  either  parallel, 
or  perpendicular  to  the  frame.  Opposite  to  this  mirror,  in  most 
sextants,  there  is  attached  permanently  to  the  frame  a  tele- 
scope, while  in  others  a  small  hole  in  a  screen  at  L  is  substi- 
tuted for  it. 


34 


EXPERIMENTAL   PHYSICS. 


Experiment.  —  The  sextant  is  used  to  measure  angles  sub- 
tended at  the  eye  by  two  distant  objects.  Before  taking 
a  measurement  it  is  necessary  to  know  that  the  two  mirrors  are 
parallel,  and  this  is  arrived  at  by  first  fixing  the  sliding  arm  at 


Fig   25. 

zero,  and  then  looking  through  the  telescope  and  the  unsilvered 
glass  M'  at  some  very  distant  object. 

If  the  two  mirrors  are  parallel,  the  observer  will  notice  that 

immediately  below  the  portion  of  the  object  viewed  directly  he 

2  will  see  an  image   of  the 

other  part  by  means  of 
rays  of  light  coming  from 
it  after  having  been  twice 
reflected  at  the  mirrors  M 
and  M'.  This  will  be  evi- 
dent from  Fig.  25,  in  which 
AB  and  DC  are  rays  coming 
from  a  very  distant  object. 
If  this  is  not  the  case,  then 
the  mirrors  are  not  parallel, 
V\v  but  are  made  so  by  turn- 

F.    26  ^  ing  the  mirror  M'  by  the 

screws  mentioned  previ- 
ously. This  done,  hold  the  sextant  so  that  its  plane  may  pass 
through  both  the  objects  whose  angular  distance  is  to  be 


THE   SEXTANT. 


35 


measured  ;  then  look  through  the  telescope  and  the  unsilvered 
glass  at  the  object  to  the  left,  and  turn  the  arm  EF  until  the 
eye  sees  the  image  of  the  second  object,  after  the  rays  forming 
it  have  been  twice  reflected,  coincide  with  the  object  seen 
directly.  The  angular  distance  between  the  objects  is  then 
twice  the  angle  between  the  mirrors,  and  therefore  twice  the 
angle  through  which  the  movable  arm  has  been  turned,  as  is 
evident  from  the  following  proof. 

In  Fig.  26  D  and  F  are  two  distant  objects,  BA  and  //"are 
the  two  mirrors  M  and  M',  and  DC  and  FH  are  rays  of  light 
coming  from  the  objects  D  and  F  respectively. 


The  angle 


=  the  angle  DCff-the  angle  NHC 
=  2  (angle  ECH-  angle  GHC] 
=  2  (angle  ECH+  90°  -angle  GtfC-go0) 
=  2  (angle  ACH-  angle  CHM) 
=  twice  angle  HMC,  which  is  the  angle  be- 
tween the  mirrors. 


Besides  measuring  the  altitude  of  the  sun,  the  angular  dis- 
tance between  two  stars,  or  that  between  two  very  distant 
objects,  a  very  instructive  a 

exercise  with  the  sextant 
is  to  measure  the  angle 
subtended  at  the  eye  by 
the  sun's  or  the  moon's 
disc  in  various  positions 
in  the  heavens.  In  meas- 
uring the  sun's  diameter, 
dark  glasses  may  be  placed 
in  front  of  the  mirrors  M 
and  M'.  The  sextant  may 

also  be  used   to  find   the    height  of   a   tower   whose   base    is 
supposed  inaccessible. 


36  EXPERIMENTAL   PHYSICS. 

In  Fig.  27  let  AB  represent  the  tower,  and  C  a  point  on  it  in 
the  same  horizontal  line  as  the  observer's  eye.     Then   since 

cotAEC=  — ,  and  cot  ADC=^, 
cot  ABC- cot  ADC=  —~  DC 


.:  AC= 


AC 
ED 


cot  AEC- cot  ADC 


So  that  if  the  observer,  while  in  the  position  EF,  measures 
the  angle  ABC,  subtended  at  the  eye  by  AC,  and  then  moves 
up  a  known  distance  ED,  and  observes  the  angle  ADC,  he 
can,  by  applying  the  above  formula,  find  the  length  AC,  and 
adding  to  this  the  height  of  the  observer,  that  of  the  tower  may 
be  obtained.  The  above  formula  may  be  easily  adapted  to 
logarithmic  calculation. 


XVI.    THE   GONIOMETER. 

Figure  28  shews  this  instrument,  which  is  adapted  to  measure 
the  angle  of  a  crystal.     It  consists  of  a  divided  circle,  which 
has  at  its    center  a  small  glass  plat- 
form concentric  with   it,  and   capable 
of  an  independent  rotation. 

The  instrument  is  provided  with  a 
collimator,  at  one  end  of  which  is  an 
adjustable  slit  which  can  be  illumi- 
nated, and  at  the  other  there  is  inserted 
a  convex  lens  to  cause  the  rays  coming 
from  the  slit  to  issue  parallel  to  each 
other.  There  is  also  a  telescope,  at- 
tached  to  a  sliding  arm,  that  can  be 
rotated  about  the  axis  of  the  divided  circle.  To  the  arms  of 
the  small  glass  platform,  and  the  telescope  are  attached  verniers 
which  generally  permit  readings  to  be  taken  to  minutes. 


THE   GONIOMETER.  37 


METHOD  I. 

Theory.  —  In  Fig.  29  AD  is  a  beam  of  parallel  light,  which 
on  striking  the  angle  EDF  of  the  crystal  at  right  angles  to  the 
edge  is  broken  up  into  two  beams  which  are  reflected  along  DB 
and  DG. 

By  the  laws  of  reflection,  the  beam  BD  has  been  deviated 
from  its  original  path  AD  through  an  angle  equal  to  twice  the 
angle  between  AD  and  ED  produced,  and  also  the  beam  DG 
through  twice  the  angle  between  AD  and  FD  produced. 
Therefore  the  angle  between  BD  and  GD  is  equal  to  twice  the 
sum  of  the  angles  made  by  ED  and  FD  produced,  with  AD, 
which  is  twice  the  angle  of  the  crystal. 

Experiment.  —  Place  the  crystal  upon  the  central  glass  plat- 
form so  that  its  edge  is  in  the  axis  of  the  graduated  circle, 
and  so  turned  that  the  light  from  the  illuminated  slit,  after 
passing  through  the  collimator,  strikes  upon  this  edge,  and 
being  split  up  into  two  beams  is  reflected  from  each  of  the  two 
faces  forming  the  angle.  This  may 
be  accomplished  by  trial  with  the 
telescope.  Turn  the  telescope  so 
as  to  intercept  one  of  these  rays, 
and  note  the  reading  on  the 
divided  circle ;  then  turn  it  until 
it  receives  the  other,  reflected 
beam,  and  again  note  the  reading. 
The  difference  between  these  two 
readings  is  twice  the  angle  of  the 
crystal.  From  the  theory  the 

student  will  readily  see  that  great  care  must  be  taken,  if  reliable 
work  is  desired,  to  have  the  edge  of  the  crystal  placed  accurately 
over  the  axis  of  the  circle,  and  perpendicular  to  it.  This  condi- 
tion of  perpendicularity  may  be  obtained  by  noting  when  the 
edge  of  the  crystal  is  in  line  with  its  image  seen  reflected  in  the 
glass  platform. 


38  EXPERIMENTAL   PHYSICS. 

It  is  also  desirable  to  choose  a  part  of  the  edge  for  the  light 
to  fall  on  which  has  the  smallest  number  of  flaws  in  it,  or  else 
it  will  be  difficult  to  take  the  telescope  readings  with  accuracy. 


METHOD  II. 

Theory.  —  In  Fig.  30  AB  is  a  beam  of  parallel  light  incident 
upon  the  edge  of  the  crystal  DBE  at  the  angle  B,  BC  is  the 
path  taken  by  this  beam  when  reflected  by  the  face  BE,  and 
D'BE'  is  the  same  crystal  in  its  second  position,  with  its  face 
D'B  now  in  the  plane  formerly  occupied  by  BE'.  From  this 
it  is  evident  that  the  ray  on  striking  the  face  D'B  will  be 
again  reflected  along  BC. 

Now  since  BE  is   in  the  same  straight   line  as  D'B,  it    is 

clear  that,  as  DB  in  the  first 
position  corresponds  with 
D'B  in  the  second,  the  crys- 
tal has  been  turned  through 
an  angle  equal  to  180°,  less 
^===<7  the  angle  DBE,  so  that 
if  we  measure  the  angle 
through  which  the  crystal 
is  turned,  and  subtract  this 
from  two  right  angles,  we 
get  the  angle  of  the  crystal. 

Fig.   30. 

Experiment.  —  Place  the  crystal  with  its  edge  as  indicated 
in  the  previous  experiment ;  then  turn  the  telescope  until  it 
receives  the  ray  BC,  and  clamp  it  there.  Then  note  the  read- 
ing on  the  divided  circle  as  indicated  by  the  vernier  on  the  arm 
attached  to  the  platform  on  which  the  crystal  rests.  After  this 
turn  this  arm  until,  on  looking  through  the  telescope,  the  ob- 
server again  sees  the  light  reflected  along  BC.  This  indicates 
that  the  crystal  is  in  the  second  position  D'BE',  and  by  again 
noting  the  reading  on  the  divided  circle  the  angle  through 


RADIUS   OF   CURVATURE.  39 

which  the  crystal  has  been  turned  may  be  found.  To  secure 
accurate  results  the  same  precautions  must  be  taken  as  in 
finding  the  angle  by  the  previous  method. 


XVII.     RADIUS   OF   CURVATURE    OF   A   CONCAVE   SPHERICAL 
MIRROR. 

Theory.  — In  Fig.  31  DHE  is  a  section  of  a  concave  spheri- 
cal mirror  whose  center  is  C.     P  is  the  illuminated  point,  P' 


Fig.  31. 

its  image,  and  PA,  AG  are  the  incident  and  reflected  rays. 
By  the  law  of  reflection,  the  angle  PAC  is  equal  to  the  angle 
CAG,  and 

sinPAC^smP'AC       .    PC  =P'C 
'  smACP~sinACP''       '  AP     AP1' 

Now  limit  the  ray  PA  by  supposing  it  incident  close  to  H, 
the  point  where  PC  produced  meets  the  mirror.  Then  AP  and 
AP'  will  become  equal  to  HP  and  HP'  respectively, 

.   PC      P'C 


'  '  PH     P'H 
PH-HC     HC-P'H 


PH  P'H 


I  I  2 


40  EXPERIMENTAL   PHYSICS. 

or,  denoting  PHt  P'H,  and  HC  by  /,  /',  and  r  respectively,  we 

obtain  the  formula  -  -\ — ;  =  -,  connecting  the  distances  of  the 
p    p1     r 

image  and  the  object  from  the  mirror  with  the  radius  of  cur- 
vature. 

Experiment.  —  Place  an  illuminated  object  before  the  mirror, 
and  receive  its  image  on  a  screen  situated  so  as  not  to  cut  off 
all  the  rays  issuing  from  P.  Then  measure  /  and  p'  directly, 

and  substituting  their  values  in  the  formula  -H — -  =  -,  obtain 

p     p       r 

the  value  of  r.  The  experiment  may  be  varied  by  giving  p 
values  ranging  from  infinity  to  -,  noting  particularly  the  case 
where  p  =  r,  in  which  case  p'  also  equals  r,  and  image  and 
object  coincide.  For  values  of  /  less  than  -,  it  is  evident  from 

the  formula  that  p'  is  negative.  The  physical  signification  of 
this  is  that  the  image  is  then  virtual,  and  so  can  no  longer  be 
received  on  a  screen. 

For  an  illuminated  object,  a  candle  or  small  gas  flame  may  be 
used,  or,  better  still,  a  fine  platinum  wire  suspended  in  a  Bun- 
sen  flame,  or  else  a  dark  thread  stretched  across  a  small  illu- 
minated opening  in  a  screen. 

For  ordinary  purposes  it  will  be  sufficient  to  make  the 
measurements  with  a  millimeter  scale,  but,  for  greater  accu- 
racy, place  the  illuminated  object  and  the  mirror  on  stands 
which  slide  freely  on  a  bench  with  scale  attached.  By 
using  verniers  on  these  stands  very  close  results  can  be 
obtained.  A  very  useful  exercise  in  connection  with  this 
experiment  is  the  determination  of  the  radii  of  curvature  of 
small  mirrors,  such  as  are  used  in  reflecting  telescopes,  or 
galvanometers. 


RADIUS   OF   CURVATURE. 


XVIII.  .RADIUS   OF   CURVATURE    OF   A   CONVEX    SPHERICAL 
MIRROR. 

Theory.  —  In  Fig.  32  ABA'  is  a  section  of  a  convex  spherical 
mirror  whose  center  is  C.    P  is  the  position  of  a  lighted  candle, 


Fig.  32. 


or   gas   jet,  P'  its  image,  and  PA,  AN  are  the  incident  and 
reflected  rays. 

By  the  law  of  reflection,  the  angle  MAL  is  equal  to  the  angle 
MAN,  and  therefore  also  equal  to  the  angle  P1  AC. 


smPAC=smP'AC 
sin  ACP~  sin 


PC  =  P'C 
PA      P'A 


Now  limit  the  ray  PA  by  supposing  it  incident  close  to  B, 
the  point  where  PC  cuts  the  mirror.  Then  PA  and  P'A  will 
become  PB  and  P'B  respectively,  and 

PC  =  P'C 
'  PB     P'B 

PB+BC  ^BC-P'B 
PB  P'B 


42  EXPERhMENTAL   PHYSICS. 

or,  denoting  PB,  P'  B,  and  BC  by  /,  />',  and  r  respectively,  we 
have  for  a  formula  connecting  the  distances  of  the  object  and 
the  image  from  the  mirror,  with  the  length  of  the  radius, 


Experiment.  —  Place  the  mirror  in  an  upright  position.  Then 
cover  the  face  of  it  with  a  sheet  of  paper  having  two  holes  in  it, 
denoted  by  A  and  A',  at  equal  distances  from  the  center  of  the 
surface  of  the  mirror.  It  will  probably  also  be  found  most  con- 
venient to  have  them  in  the  same  horizontal  plane  as  the  center. 
By  this  arrangement  there  will  be  two  rays  of  light,  AN  and 
A'N',  reflected  from  the  mirror.  Intercept  these  rays  by  a 
screen  NN'  so  made  as  to  permit  this,  and  yet  not  cut  off  all 
the  light  coming  to  the  mirror.  The  measurements  then  to  be 
taken  are  PB,  the  distance  between  the  candle  and  the  mirror, 
BT  that  of  the  screen  from  the  mirror,  AA'  that  between  the 
holes  in  the  paper,  and  NN1  that  between  the  spots  of  light 
formed  by  the  intercepted  rays  on  the  screen. 

Then,  since  the  triangles  P'AG  and  P'NT  are  similar,  we 


have  =  PB^BTj  or  if  A  A1  is  very  small, 

P'B  =  P'B  +  BT 

AB~       TN 

p,B=AB-BT  =  AA'-BT 

TN-AB     NN'-AA' 

Therefore  P'B  or  /'  can  be  found,  and  having  measured 
BP,  or  p  directly,  we  can,  by  substituting  the  values  of  these 

two  quantities  in  the  formula   ---  =  -,  obtain  ;'. 

/     /      r 

In  order  to  obtain  accurate  results  in  this  experiment,  the 
two  holes  in  the  paper  must  be  taken  as  close  -together  as  cir- 
cumstances will  permit  ;  and,  as  it  is  somewhat  difficult  to  meas- 
ure accurately  the  distance  between  the  spots  of  light,  it  will  be 


RADIUS   OF   CURVATURE. 


43 


well  to  have  the  holes  in  the  paper  very  small,  and  to  take 
great  care  to  have  their  edges  neatly  cut. 

Note  on  the  construction  of  images.  —  By  an  extension  of  the 
theory  given  in  the  two  preceding  experiments,  we  may  find  the 
position  of  the  image 
of  an  object  which  is 
not  a  point,  as  we  there 
supposed  it  to  be.  In 
Fig.  31  let  PQ  be  the 
object.  Join  QC,  and 
produce  it  to  meet  the 
section  DHE  in  B. 
Then  just  as  we  found 
P',  the  image  of  P, 
to  lie  on  PH,  and  to  be 
given  by  the  formula  


Fig.  33. 


must  lie  on  QB  and  be  given  by  the  relation 1 — ?_=_?_. 

QB     QB    BC 

By  joining  Q'  and  P1  as  thus  found,  Q'P'  will  be  the  image  of 
QP.  From  the  above  explanation  it  will  be  readily  seen  that 
Q'P',  will  not  be  quite  perpendicular  to  either  PH,  or  QB,  and 
it  is  therefore  for  this  reason  that  objects  as  seen  in  a  spherical 

mirror   appear   distorted.       From  the  formula  -H —  =  -,  it  will 

p    p'     r 

be  seen  that  if  the  object  be  placed  off  at  an  infinite  distance 
the  image  will  be  at  a  distance  -  from  the  mirror.  This  point 

is  called  the  focus,  and  as  all  rays  coming  to  the  mirror  from  a 
point  at  an  infinite  distance  are  practically  parallel,  we  may 
define  it  to  be  the  point  through  which  pass  all  the  reflected 
rays  which  are  incident  on  the  mirror  parallel  to  the  axis,  PH. 
Thus  we  see  the  ray  QA,  which  is  parallel  to  PH,  will, 
when  reflected,  approximately  pass  through  the  focus  F  which 
bisects  CH.  Again,  a  ray  QC  which  passes  through  the  center 
will,  when  reflected,  come  back  on  itself.  The  point  where  this 


44 


EXPERIMENTAL   PHYSICS. 


ray  intersects  the  ray  AF  will  be  the  image  of  Q,  and  by  drop- 
ping a  perpendicular  from  this  point  on  the  axis  PH,  we  will 

approximately  have  the 
position  of  P'Q'  the 
image  of  PQ. 

This  method  of  con- 
structing images  is  ex- 
hibited in  Figs.  31,  33, 
and  34,  and  it  will  be 
an  instructive  exercise 
for  the  student  to  apply 
the  same  method  to  the 


Fig.  34. 


case  of  a  convex  mirror. 


XIX.     RADIUS   OF   CURVATURE   OF   SPHERICAL   MIRRORS. 
GENERAL  METHOD. 

Theory.  —  In  Fig.  35  HDG  is  a  section  of  a  convex  mirror 
whose  center  is  C;  A  and  B  are  two  illuminated  objects;  A  'and 
B'  are  their  images;  EF  is  a  finely  divided  scale  placed  before 


G' 


Fig.  35. 

the  mirror  in  the  same  horizontal  plane  as  A  and  B;  and  PG',  a 
telescope  placed  in  the  line  joining  C  to  the  point  midway  be- 
tween the  objects  A  and  B,  has  its  objective  in  the  line  AB. 
L  and  L'  are  the  points  where  the  lines  A'P  and  B'P  cut  EF. 


RADIUS   OF   CURVATURE.  45 

The  formula  connecting  image  and  object  for  convex  mirrors 

is  -=- — -;  and  since  A',  B',  and  P'  are  the  images  of  A,  B, 
p    p'     r 

and  P  respectively,  we  have  denoting  LL'  by  y,  AB  by  x,  A'B' 
by  x',  PD  by/,  P'D  by/',  and  PP'  by/, 

rH!  (I> 

and  by  comparing  the  sizes  of  the  image  and  of  the  object,  we 
have 

j-TJp-  & 

From  equation  (i)  we  have 


/     r    /      r 

r~T=~pr' 

•  P'  =  r-P'  =  CP>  =*', 
'  p      r+p      CP      x 


Again,  from  (3),         /'  =  — z — >  (4) 

r+2/ 


r+2p 


(5) 


From  the  figure         —  =  y—. 

y    P 


V  =  P-     xr    ,  from  (5) 

/  r+2p 

^-± — -  from  (4) 


2(r+p) 

:-?£L  (6) 

x—2y 


46  EXPERIMENTAL   PHYSICS. 

By  a  process  of  reasoning  similar  to  this,  the  student  may 
easily  find  for  himself  that  in  the  case  of  concave  mirrors  the 
formula  becomes 

-    2^  (7) 


Experiment.  —  If  this  experiment  is  performed  in  the  dark, 
two  lighted  candles,  or  gas  jets  may  be  used  for  the  illuminated 
points  A  and  B ;  and  in  case  the  scale  FE  is  not  sufficiently 
illuminated  to  be  easily  read,  it  may  be  made  so  by  projecting 
a  beam  of  light  on  it  by  means  of  a  convex  lens.  However, 
it  is  better  to  perform  the  experiment  in  a  well-lighted  room, 
and  to  use  two  porcelain,  or  ivory  scales  with  divisions  made 
in  black,  the  one  to  take  the  position  FE,  and  two  marks  on 
the  other  to  replace  the  illuminated  points  A  and  B.  Having 
then  placed  the  mirror,  telescope,  illuminated  objects,  and  scales 
in  the  positions  indicated,  adjust  the  telescope  so  as  to  be  able 
to  read  the  scale  FE.  If  PD  is  taken  of  considerable  length, 
A1  and  B'  will  be  approximately  in  focus  at  the  same  time  as 
FE,  and  the  length  LV  on  the  scale,  cut  off  by  the  space 
between  A'  and  B',  may  thus  be  directly  read  off  by  means  of 
the  telescope.  The  lengths  p  and  x  are  measured  with  ordinary 
scales,  and  for  accuracy  it  would  be  well  to  check  the  work  by 
giving  these  quantities  different  values. 

This  method  may  also  be  used  to  find  the  radius  of  curvature 
of  the  surface  of  a  lens.  In  this  case  each  face  of  the  lens 
will  give  a  pair  of  images.  By  noting  that  the  images  produced 
by  a  convex  reflecting  surface  are  always  erect,  the  student  will 
find  no  difficulty  in  distinguishing  the  two  sets. 


FOCAL  LENGTH  OF  A  BICONVEX  LENS.       47 

XX.  FOCAL  LENGTH  OF  A  BICONVEX  LENS. 
METHOD  I. 

Theory. — In  Fig.  36,  AC  is  a  biconvex  lens  whose  focus  is 
at  F,  PP'  is  the  principal  axis,  PQ  is  an  illuminated  object,  and 
P'Q'  its  image.  If  we  assume  that  a  ray  of  light  striking  the 
lens  at  its  center  passes  through  without  deviation,  and  that 


Fig.  36. 

a  ray  parallel  to  the  principal  axis  on  striking  it  ultimately 
passes  through  the  focus,  we  are  able  to  obtain  a  relation 
connecting  the  distance  of  the  object,  and  of  its  image  from 
the  center  of  the  lens  C,  with  the  focal  length. 

If  QC  and  QA  are  these  two  rays,  the  point  Q'  where  they 
intersect  will  be  the  image  of  Q,  and  dropping  a  perpendicular 
Q'P'  on  PP',  P'Q'  will  be  the  image  of  PQ. 

Denoting  CP  by  p,  CP'  by  /,  and  CF  by  /,  we  have,  from 
similar  triangles, 


P'Q'~  S' 

Again,  by  erecting  at  C  a  line  perpendicular  to  PP',  and  ter- 
minated at  the  point  where  it  cuts  the  ray  QA,  we  have 

(2) 


.-.  from  (i)  and  (2) 

P'-f  _  PL     .  -I  +  1--L 

~r~  p   -  p'  P  / 


48 


EXPERIMENTAL   PHYSICS. 


Experiment.  —  Having  placed  the  object  and  the  lens  in  the 
same  horizontal  line,  move  the  screen  until  the  image  is  dis- 
tinctly focussed  on  it.  Then  measure  directly  the  distance  of 
the  image  and  of  the  object  from  the  center  of  the  lens,  i.e. 
p  and  /',  and  substituting  the  values  of  these  quantities  in  the 

formula  —  -\ — =  — ,  obtain  f,  the  focal  length.    As  in  the  case 
of  concave  mirrors,  although  a  lighted  candle,  or  gas  jet  may 


Fig.  37. 

be  taken  for  the  illuminated  object,  it  will  be  found  that  more 
accurate  results  are  obtained  when  an  incandescent  platinum 
wire,  or  a  dark  thread  stretched  over  an  illuminated  opening 
in  a  screen  is  used.  If  this  experiment  is  conducted  in  a  dark 
room,  it  will  be  found  that  a  screen  painted  white  or  one 
made  of  ground  glass  will  be  the  most  suitable.  Great  care 
should  be  taken  to  have  the  image  well  denned. 

If  the  object  be  at  a  great  distance,  the  formula  indicates 
that  the  image  will  then  be  at  the  focus,  and  the  focal  length 
may  then  be  measured  directly.  If/  be  taken  equal  to  /',  each 
is  then  equal  to  2/,  and  in  that  case,  the  distance  between 
object  and  screen  need  only  be  measured.  If  /  be  taken  less 
than  /,  the  formula  indicates  that  /'  becomes  negative,  which 
signifies  that  the  image  is  then  virtual.  This  is  the  case 
when  a  convex  lens  is  used  as  a  simple  microscope,  and  Fig. 
37  then  indicates  the  relative  positions  of  the  image  and  the 
object. 


FOCAL   LENGTH   OF   A   BICONVEX   LENS.  49 


METHOD  II. 

In  Fig.  38  P  is  an  illuminated  object,  L  a  biconvex  lens,  and 
P'  the  position  of  a  screen  with  the  image  of  P  clearly  defined 
on  it.  From  the  symmetry  of  this  arrangement,  as  well  as  from 

L  L' 


Fig.  38. 

an  inspection  of  the  formula  -+-  =  -,  it  can  be  readily  seen 

P     P    f 

that  if  the  lens  be  moved  from  L  to  L1  so  that  P'L'  is  equal  to 
PL,  there  will  again  be  an  image  of  P  on  the  screen  at  P'. 
Denoting  then  the  lengths  PP'  and  LL'  by  /  and  a  respectively, 
we  have 


V  ~~A     *          ~ 

p  —  -  and  #  =  -- 
2  2 

Substituting  these  values  of  /  and  /'  in  the  formula 

/47=7 

we  have 

•'•          /-^  '     <» 

Experiment.  —  Place  the  illuminated  object  and  the  lens  in 
position  at  some  chosen  distance  apart,  which,  as  is  evident 
from  Fig.  37,  must  be  greater  than  the  focal  length  of  the  lens. 
Then  adjust  the  screen  so  that  the  image  of  P  is  quite  distinct 
on  it,  and  measure  PP',  or  /  the  distance  between  the  object 
and  the  image.  Measure  also  the  distance  a  through  which 
the  lens  must  be  moved  in  order  to  have  an  image  of  P  focussed 
on  the  screen  a  second  time,  and  substituting  these  values  of 
/  and  a,  in  formula  (i),  find  /  the  focal  length  of  the  lens. 


50  EXPERIMENTAL   PHYSICS. 

This  method  is  especially  applicable  in  finding  the  focal 
lengths  of  combinations  of  lenses  such  as  the  objectives  and 
eyepieces  of  optical  instruments,  or  those  used  in  photographic 
cameras. 

NOTE.  —  If  non-achromatized  lenses  are  used  in  these  experi- 
ments, it  is  generally  difficult  for  a  teginner  to  decide  where 
the  screen  should  be  really  placed. 

In  Fig.  39  AB  represents  a  simple  non-achromatized  lens, 
P  an  illuminated  point,  and  PA  and  PB  the  limiting  rays 
of  white  light  which  strike  the  lens.  These,  when  refracted 


Fig.  39. 

through  the  lens,  become  dispersed,  and  two  pencils  of  coloured 
rays,  DAC&nd  FBE,  are  obtained.  AD  and  BE  will  therefore 
be  red  rays,  and  AC  and  BF  violet,  since  the  latter  are  more 
refracted  than  the  former.  It  will  be  readily  seen  from  the 
figure  that  any  point  between  G  and  H  may  be  said  to  be  the 
image  of  P.  If  the  screen  be  placed  at  G,  the  image  on  it 
will  be  fringed  with  a  reddish  tint,  while  if  placed  at  H  it  will 
have  a  violet  border.  Between  these  two  points  then  there 
will  be  a  point  K  where  the  red  and  the  violet  rays  overlap,  and 
the  image  will  be  of  one  colour.  The  screen  should  therefore 
be  placed  at  this  point. 


FOCAL  LENGTH  OF  A  BICONCAVE  LENS.       51 

XXI.  FOCAL  LENGTH  OF  A  BICONCAVE  LENS. 
METHOD  I. 

Theory In  Fig.  4.0  AC  is  a  biconcave  lens  whose  focus  is 

at  F,  PP'  is  the  principal  axis,  PQ  is  an  illuminated  object,  and 
P'Q'  is  its  image.  If  we  assume,  as  in  the  case  of  a  convex 
lens,  that  a  ray,  on  striking  the  lens  at  the  center,  passes  through 
without  deviation,  and  that  one  parallel  to  the  principal  axis 
appears  ultimately  to  diverge  from  the  focus,  we  may  also  in 


Fig.  40 

this  case  obtain  a  formula  connecting  the  distance  of  the  image 
and  of  the  object  from  the  center  of  the  lens  C,  with  its  focal 
length.  QA  and  QC  are  two  such  rays,  and  as  on  emerging 
they  appear  to  come  from  Q',  therefore  Q'f,  the  perpendicular 
from  Q'  on  PC,  will  be  approximately  the  image  of  QP.  Denot- 
ing PC  by  /,  P'C  by  p\  and  FC  by  /,  we  have  from  similar 
triangles, 


If  a  line  perpendicular  to  PP'  be  erected  at  C  to  meet  the  ray 
QD,  we  will  have  pn          f 


From  (i)  and  (2)  7='  (3) 

TT? 

which  is  the  formula  required. 


EXPERIMENTAL   PHYSICS. 


It  will  be  seen,  in  this  case,  that  the  image  is  always  virtual, 
and  so  cannot  be  measured  directly.  The  following  method  of 
finding  the  focal  length  is  generally  adopted. 

In  Fig.  41  AB  is  the  lens,  one  of  whose  surfaces  is  covered 
with  paper  pierced  by  two  small  holes,  or  slits  at  E  and  F,  which 


Fig.  41. 

permit  two  pencils  of  light  PE  and  PFto  pass  through  the  lens, 
and  to  form  two  bright  spots,  G  and  ff,  on  a  screen  placed 
behind  it.  As  these  pencils  on  issuing  from  the  lens  appear 
to  come  from  the  point  P',  it  is  therefore  the  image  of  P. 

From  similar  triangles,  GHP'  and  EFP1,  we   have  approxi- 
mately, 

GH     _EF 
P'C1 

EF-CS 


or 


P'C+CS 
P'C 


GH-EF 


(i) 


If  we  therefore  measure  the  distance  between  the  slits  E  and 
F,  and  that  between  the  spots  of  light  on  the  screen  G  and  //", 
and  also  measure  CS,  the  distance  between  the  screen  and  the 
lens,  we  may,  by  substituting  in  equation  (i),  find  P'C,  i.e.  p' . 
Having  found  PC,  or  p  directly,  we  may  then,  by  means  of  the 

formula  — =  -,  find  /"the  focal  length  of  the  lens. 

P     P    f 


FOCAL   LENGTH   OF  A   BICONCAVE   LENS. 


53 


Experiment. — The  results  in  this  experiment  may  be  checked 
by  varying  the  distance  between  the  holes,  or  slits  in  the  paper, 
that  between  the  screen  and  the  lens,  or  that  between  the  illu- 
minated point  and  the  lens.  A  lighted  candle,  or  gas  jet,  will 
be  sufficient  for  an  illuminated  object,  and  best  results  are 
obtained  when  the  slits  E  and  F  are  made  very  'small  and 
taken  close  together. 


METHOD  II. 

Theory. — The  focal  length  of  a  biconcave  lens  may  also  be 
found  by  using  it  in  combination  with  a  biconvex  sufficiently 
powerful  to  make  the  two  together  act  as  a  convex  lens.  In 


Fig.  42. 

Fig.  42  AB  is  a  biconvex  lens  with  center  C  and  focal  length 
FC,  A'B'  is  a  biconcave  lens  with  center  C  and  focal  length 
F'C,  PQ  is  an  illuminated  object,  P'Q'  the  image  that  would 
be  formed,  were  the  concave  lens  removed,  and  P"Q"  is  the 
image  of  PQ  that  is  formed  by  the  combination  ;  while  the 
heavy  line  joining  Q  and  Q"  is  the  actual  path  of  a  ray  through 
both  lenses.  From  the  figure  it  will  be  evident  that  Q"  is  the 
image  of  Q'  with  respect  to  the  lens  A'B',  because  the  ray  DE, 
which  takes  the  direction  EQ"  with  the  lens  A'B'  interposed, 
would  follow  the  path  EQ',  were  it  removed. 

By  the  principle  of  reversion  applicable  in  the  study  of  optics, 
if  Q"  were  an  illuminated  point,  and  Q"E  were  a  ray  of  light 


54  EXPERIMENTAL   PHYSICS 

coming  from  it,  then  this  ray,  after  passing  through  the  lens 
A'B',  would,  take  the  path  ED,  and  Q  would  be  the  image  of  Q". 

Since  —  --  =—  is  the  formula  for  a  concave  lens,  the  following 
equation  will  hold  : 


If  then  CQ'  and  CQ"  can  be  found,  /,  the  focal  length,  may  be 
determined. 

Experiment.  —  Take  for  an  illuminated  object  an  incandescent 
platinum  wire  ;  then  place  the  convex  lens  at  a  certain  distance 
from  it,  which  is  to  remain  the  same  throughout  the  experiment. 
On  a  screen  placed  at  Q'  receive  P'Q  ',  the  image  of  PQ  formed 
by  the  single  lens.  Measure  CQ',  and  after  removing  the  screen 
interpose  in  the  path  of  the  rays  the  concave  lens  A'B',  and 
again  receive  on  the  screen  P"Q'',  the  image  of  PQ  formed  by 
the  combination.  Measure  C'Q"  and  the  distance  between  the 
centers  of  the  lenses  CO.  Then  since  CQ'=CQ'-CC,  we  can 
obtain  CQ',  and  having  measured  C'Q"  directly,  we  may,  by  sub- 

stituting these  values  in  the  equation  777^  —  7^  —  ^  =  —  ,  obtain/, 

C'Q      C  Q      f 
the  focal  length  of  the  lens  A'B'. 

The  results  obtained  may  be  checked  by  varying  the  distance 
between  the  lenses,  or  that  of  the  incandescent  wire  from  the 
convex  lens.  In  this  experiment  the  student  may  meet  with 
some  difficulty  in  deciding  where  the  screen  should  be  placed, 
owing  to  the  lenses  not  being  achromatic;  but  by  noting  the 
precautions  given  in  Experiment  XX.  this  may  be  easily  over- 
come. 

XXII.     INDICES   OF   REFRACTION. 

Theory  --  When  a  small  pencil  of  white  light  strikes  a  prism, 
as  SE  in  Fig.  43,  it  is,  owing  to  the  unequal  refrangibility  of 
the  rays  composing  it,  broken  up  into  a  series  of  them,  each 


INDICES   OF   REFRACTION.  55 

corresponding  to  a  different  colour.  In  this  experiment,  as  we 
wish  to  deal  only  with  a  single  ray,  suppose  S£  to  be  a  simple 
ray  obtained  from  a  source  of 
monochromatic  light. 

When  the  ray  S£  strikes  the 
prism  at  E,  it  is  refracted,  and 
passing  through  the  prism 
along  EF  is  again  refracted 
at  F,  and  emerging  takes  the 

direction  FR.  EO  and  FO  are  normals  to  the  faces  of  the 
prism  at  E  and  F  respectively.  Designating  the  angles  SEG 
and  RFH  by  i  and  /',  and  the  angles  FEO  and  EFO  by  r 
and  r1,  we  have,  from  the  law  of  refraction, 

sin  z'=/i  sinr,  (i) 

sin  i'=  /j.  sin  r1.  (2) 

Also,  since  the  angle  EOF  is  the  supplement  of  the  angles 
r  and  /,  we  have 

r+r'=A.  (3) 

Again,  denoting  the  angle  of  deviation  by  D,  we  have 
D=DEF+DFE 


from  (3)  =i+i'-A.  (4) 

From  (i),  we  have 

sinz=/A  sin  r\ 
.'.  from  (3)  we  have 

sin  z  =  /4sin  (A  —  r')=/x  sin  A  cos  r1  —  p  cos  A  sin  ^ 
=  jj.  sin  A  cos  r'  —  cos  A  sin  i\ 
.-.   {sinz  +  cos^  sinz'|2  =  /*2sin2^  cos2r', 
.-.   {sinz  +  cos^  sin  i  '|2  +  sin2  A  sin2*' 

=  /i2  sin2  A  cos2  /  +  /*2  sin2  ^  sin2  / 


56  EXPERIMENTAL   PHYSICS. 

from  (2) ; 

.;.  sin2  2  +  2  cos  A  sin  i  sin  r +  sin2  z'=/n2  sin2A 

.  9  .  ,   .  o  •,   I  —  COS  21  ,  I  —  COS  2  t' 

Now  snr  z -J- sm^  z  = 1 

2  2 

—  I  — ^  {COS  2  Z  +  COS  2  Z'} 
=  I  —  COS  (z :  —  z')  COS  (z  +  Z*), 

and  since          cos  (z  —  z'')  —  cos  (i+  z')  =  2  sin  z  sin  *', 
.  •.  /i2  sin2  ^  =  i  -  cos  (z  -  z ')  cos  (z  +  z ') 

+  cos  A  [cos  (z  —  z')  —  cos  (z  +  z')]  ; 
.  •.  since  sin2  A  +  cos2  A  =  i , 
(fj?-  i)  sin2  ^4  =  [cos  ^4  +cos  (z  -  z-f) }  {cos  A  -cos  (z  +  z') } ; 

i.e.  the  product  of  the  two  quantities  on  the  right-hand  side 
is  a  constant  quantity.  Now  from  (4)  we  see  that  D  is  least 
when  (z'+z')  is  least,  i.e.  when  cos(z'+z')  is  greatest,  i.e.  when 
cos^4—  cos(z'+z'f)  is  least;  and  since  this  quantity  multiplied  by 
cos^+cos(z  — z')  equals  a  constant  quantity,  therefore  when 
D  is  least,  cos^+cos(z  — z')  is  greatest,  i.e.  D  is  least  when 
cos(z  — z')  is  greatest,  i.e.  when  i—  z'=o.  When  D,  therefore, 
is  a  minimum,  z  is  equal  to  i'.  The  deviation  is  a  minimum 
when  the  ray  SE  strikes  the  prism  so  that  the  angle  of  inci- 
dence equals  the  angle  of  emergence.  Now  when  z  =  z',  we 
have  r=fj,  and  D=2t  —  A,  and  A—2r\ 

.'.  since  sin  Z=/A  sin  r, 

.    D  +  A 


If,  therefore,  we  are  able  to  get  the  angle  of  minimum  devia- 
tion D,  and  have  found  the  angle  of  the  prism  A,  the  index 
of  refraction  p  may  readily  be  obtained  from  this  formula. 


INDICES   OF    REFRACTION. 


57 


Experiment. — Place  the  prism  on  the  glass  platform  attached 
to  the  divided  circle  described  in  Experiment  XVI.  in  such 
a  way  as  to  allow  the  light  from  the  illuminated  slit  to  fall  on 
one  of  its  faces  EA  near  the  edge  A,  and  move  the  telescope 
until  it  intercepts  the  emerging  ray.  Then  it  will  be  noticed, 
if  the  prism  be  slightly  turned  in  one  direction,  that  the  angle 
of  deviation  is  increased,  while  if  it  is  turned  in  the  opposite 
direction,  this  angle  will  be  diminished.  Continue  then  to 
turn  it  in  this  latter  direction  and  following  the  displaced  ray 
with  the  telescope,  it  will  be  noticed  that  at  a  certain  instant 
the  emerging  ray  stands  still,  and  on  the  prism  being  turned 
still  further,  it  begins  to  turn  back,  and  therefore  the  angle 
of  deviation  begins  to  increase.  The  deviation,  when  the 
emerging  ray  becomes  stationary,  is  then  a  minimum,  and  the 
angle  the  incident  ray  makes  with  one  face  of  the  mirror  is 
equal  to  the  angle  the  emerging  ray  makes  with  the  other. 
When  the  emerging  ray  is  in  this  position,  note  the  reading 
on  the  divided  circle  indicated  by  the  vernier  attached  to  the 
telescope,  then  remove  the  prism,  and  turn  the  telescope  so 
as  to  receive  the  light  directly  from  the  slit,  and  again  note 
the  reading.  The  difference  between  these  two  readings  will 
be  the  angle  D  of  minimum  deviation. 

Substituting  its  value  in  the  formula 


the  angle  A  having  been  previously  measured,  p.,  the  index  of 
refraction,  may  be  found.  A  simple  device  which  will  enable 
the  slit  to  be  illuminated  with  monochromatic  light,  is  to  fill 
with  a  sodium  salt  a  glass  tube  drawn  to  a  fine  point  through 
which  passes  a  fine  platinum  wire.  If  the  tube  be  then 
placed  so  that  the  platinum  wire  is  in  the  flame  with  which 
the  slit  is  to  be  illuminated,  the  heated  wire  melts  the  salt, 


58  EXPERIMENTAL   PHYSICS. 

which  then  runs  into  the  flame  and  produces  the  yellow  sodium 
light.  In  finding  the  refractive  index  of  a  liquid,  place  it  in 
a  glass  bottle  prism,  and  proceed  just  as  indicated  for  a  solid 
prism. 


XXIII.    EXAMPLES   OF   MAGNIFICATION   WITH   LENSES. 

In  estimating  size  we  are  guided  almost  wholly  by  the  angle 
subtended  at  the  eye  by  the  object  viewed.  As  this  angle 
depends  on  the  distance  of  the  object  from  the  observer,  it  is 
necessary  to  say  where  it,  and  its  image  formed  by  a  lens  or 
system  of  lenses,  should  be  placed  in  order  that  we  may 
proceed  to  a  fit  determination  of  the  magnifying  power.  Com- 
mon experience  teaches  us  that  the  eye  can  accommodate  itself 
to  different  distances.  Persons  with  normal  sight  can  see 
distinctly  the  contour  and  general  appearance  of  far-removed 
objects,  such  as  the  sun,  the  moon,  mountains,  or  distant 
buildings,  and  can  also  see  distinctly  objects  placed  as  close  to 
the  eye  as  fifteen  centimeters.  Just  as,  in  regard  to  the  sense 
of  touch,  we  are  not  able  to  distinguish  between  two  points  of 
contact,  when  the  distance  between  them  is  less  than  a  certain 
limiting  value,  so  in  the  case  of  sight  the  eye  is  not  able  to 
distinguish  points  on  the  body  viewed,  when  the  distance 
between  them  subtends  an  angle  at  the  eye  less  than  a  certain 
definite  and  determinate  limit.  The  angle  subtended  by  two 
points  of  course  increases  as  they  are  brought  closer  to  the 
observer,  and  hence  it  is  that  the  nearer  the  object,  the  more 
detail  there  is  visible.  In  comparing,  therefore,  the  sizes  of 
objects  they  should  be  placed  where  there  is  the  most  detail 
visible, — that  is,  at  the  nearest  point  of  distinct  vision,  —  and 
in  considering  the  magnitude  of  an  image  the  object  should  be 
so  placed  with  regard  to  the  lens  that  the  image  of  it  will  appear 
to  be  at  this  distance.  The  smallest  distance  of  distinct  vision 
is,  however,  very  different  for  different  persons,  ranging  all  the 
way  from  fifteen  to  thirty  centimeters,  and  as  the  magnifying 


EXAMPLES   OF   MAGNIFICATION   WITH   LENSES.  59 

power  of  an  instrument  should  give  an  idea  of  enlargement  for 
an  eye  in  general,  it  has  been  agreed  to  take  the  distance  of 
distinct  vision  as  twenty-five  centimeters,  and  in  what  follows 
we  will  always  suppose  the  image,  as  formed  by  the  lenses, 
seen  there.  The  magnifying  power  of  an  optical  instrument  is 
generally  expressed  as  so  many  diameters,  which  signifies  that 
the  linear  dimensions  of  the  image  and  of  the  object,  and  not 
their  areas,  are  to  be  compared.  In  the  case  of  magnifying 
glasses  and  microscopes  the  enlargement  is  taken  to  be  the 
ratio  of  the  apparent  diameters  of  the  image  and  of  the  object, 
both  being  seen  at  the  distance  of  distinct  vision. 

The  Magnifying  Glass.  —  The  simplest  case  of  magnification 
is  that  produced  by  means  of  a  single  biconvex  lens.  In  Fig.  37 
let  AB  be  such  a  lens,  O  the  position  of  the  eye,  PQ  the  object 
viewed,  and  P'Q'  the  image  of  PQ  seen  at  the  distance  of 
distinct  vision,  OP'.  Denoting  then,  OP'  by  A,  OC  by  d,  CP 
by  p,  CP'  by/',  and  the  magnification  by  G,  we  have 

G_P'Q 
—' 


Now 

p'          p1 

a  11 J  since  p'  =  k—d, 

we  have  G=\-\ — =— ;  (i) 

or,  if  the  eye  is  close  to  the  lens,  G  =  i  +  -p 

In  order  to  test  this  result,  take  a  finely  divided  scale  for  the 
object,  and  adjust  the  lens  so  that  the  divisions  are  distinctly 
visible.  If  a  similar  scale  be  placed  at  the  distance  of  distinct 


6o 


EXPERIMENTAL   PHYSICS. 


vision,  and  be  viewed  with  one  eye  while  the  image  of  the 
second  one  is  viewed  with  the  other,  the  observer  will  be  able, 
after  a  little  practice,  to  tell  how  many  divisions  on  the  scale 
seen  directly  correspond  with  one  division  on  the  image.  This 
number  may  be  taken  as  a  measure  of  the  magnifying  power  of 
the  lens.  A  number  of  trials  should  be  made  by  placing  the 
eye  at  different  distances  from  the  lens,  and  in  order  to  correct 
any  error  arising  from  a  difference  in  the  eyes  of  the  observer, 
the  one  should  be  used  as  often  as  the  other  in  viewing  each  of 
the  scales. 

Doublets.  —  From  formula  (i)  it  is  evident  that  the  smaller 
we  take  f,  the  greater  does  the  magnification  become.  When, 
however,  we  increase  the  curvature  of  the  lens,  spherical  aberra- 


.#' 


Fig.  44. 

tion  increases  also,  and  in  order  to  avoid  this,  and  yet  have 
considerable  enlargement,  it  is  customary  to  use  a  combination  of 
two  lenses  separated  by  an  interval,  for  the  eyepiece  of  a  micro- 
scope or  telescope.  Such  a  combination  is  called  a  doublet, 
and  its  magnifying  power  may  be  calculated  in  the  same  manner 
as  that  of  a  single  lens. 

In  Fig.  44  let  AB  and  A'B'  be  the  two  lenses  composing  the 


EXAMPLES   OF   MAGNIFICATION   WITH   LENSES.         6l 

doublet.  Let  f  and/'  be  their  focal  lengths  respectively,  and 
let  PQ  be  the  object  viewed,  PiQ1  its  image  formed  by  the 
lens  AJ3,  and  PZQ2  the  final  image  formed  by  the  combination, 
and  seen  at  the  distance  of  distinct  vision  A,  by  an  eye  placed 
close  to  the  lens  A'B1.  Denoting  then  CQ  by/,  CQl  by  /,  and 
CO  by  D,  we  have 

H—> 

and 

1          '  ' 


A     /  +  £>         /' 
The  magnification,  which  is  denoted  by  G,  is  given  by 


G~~PQ~P&l'~PQ' 
i.e. 

G-^n'f-- 
p  +£>    p 

Now  from  (3) =  I  H — f,  and  from  (2)  and  (3) 

/=  A/'         D 

p        h  /(A  +  /)     /' 


(3) 


When  the  two  lenses  have  the  same  focal  length,  and  the 
interval  between  them  is  equal  to  two-thirds  of  this  length,  the 
doublet  is  called  a  Ramsden  eyepiece.  Its  magnifying  power 

is  given  by  G  =  -  +  ^-—  •     In  the  Wollaston  Doublet,  /'  =  3/,  D 

3     3  / 
is  made  equal  to  -|/,  and  its  magnifying  power  is  therefore 

5A_I 
6/     2 

The  combination  most  frequently  used  as  the  eyepiece  for 
microscopes  is  that  of  Huyghens.  It  is  sometimes  called  a 


62  EXPERIMENTAL    PHYSICS. 

negative  eyepiece,  to  distinguish  it  from  a  Ramsden  or  positive 
eyepiece.  In  it/'  =  J/,  D  is  taken  equal  to  f /,  and  its  magni- 
fying power  is  given  by  G=-  +  2 — .  When  a  doublet  is  used  as 

an  eyepiece,  the  lens  next  the  objective  is  called  the  field  glass, 
and  that  next  the  eye  the  eyeglass. 

In  testing  the  magnification  of  a  doublet  experimentally, 
exactly  the  same  method  is  adopted  as  in  the  case  of  a  single 
lens.  If  the  eye  is  not  placed  close  to  the  lens  in  doing  this,  an 
allowance  must  be  made  in  formula  (4)  for  its  distance  from  it, 
similar  to  that  for  enlargement  by  a  magnifying  glass. 

The  Compound  Microscope.  —  In  its  simplest  form  the  com- 
pound microscope  consists  of  two  condensing  lenses,  one  of 
which  is  called  an  object  glass,  or  objective,  and  the  other 
an  eyeglass,  or  eyepiece.  The  objective,  however,  is  usually 
made  up  of  a  system  of  lenses  so  constructed  as  to  reduce 
chromatic  and  spherical  aberration  to  a  minimum,  and  the 
eyepiece  most  generally  used  is  that  of  Huyghens.  Just  as 
in  the  case  of  doubtlets,  an  expression  for  magnification,  by  a 
microscope  can  be  calculated  in  terms  of  the  focal  lengths  of 
the  lenses,  the  distances  these  are  apart,  and  the  distance  of 
distinct  vision.  Experimentally  the  same  method  as  that  just 
described  for  a  magnifying  glass  may  be  adopted  ;  but  it  is 
usual  in  the  case  of  the  microscope  to  modify  it  by  placing  a 
camera  lucida  over  the  eyepiece,  and  by  placing  a  scale  a  little 
to  one  side  of  the  instrument,  so  that,  with  the  same  eye  the 
image  of  the  scale  under  the  microscope  and  the  reflected 
image  of  the  scale  at  the  side  are  both  visible  at  once.  By 
noting  how  many  divisions  on  the  image  formed  by  reflection 
correspond  with  one  on  that  formed  by  refraction  the  magni- 
fying power  can  be  ascertained. 

The  microscope  is  frequently  used  for  measuring  small 
distances.  A  reference  to  Fig.  45  will  explain  how  this  is 
accomplished. 

Here  AB  represents  the  objective,  Al^l  and  A^B^  the  field 


EXAMPLES   OF   MAGNIFICATION   WITH   LENSES.          63 

glass  and  eyeglass  respectively  of  an  Huyghens  eyepiece ;  PQ 
the  object,  P\Q\  the  image  the  rays  go  to  form  after  passing 
through  the  objective,  P2Q2  the  real  image  formed  by  this  lens, 
and  the  field  glass,  and  PSQ3  the  final  image  seen  at  the  dis- 
tance of  distinct  vision.  A  micrometer  EF  (consisting  of  a 
thin  glass  plate  on  which  there  is  generally  ruled  a  half-centi- 
meter divided  into  fiftieths)  is  placed  in  the  eyepiece  in  such 
a  position  that  its  image  seen  through  the  eye  lens  is  at  the 
distance  of  distinct  vision.  When,  then,  an  object  PQ  is 
viewed,  and  its  final  image  PSQ3  is  seen  distinctly,  the  real 


Fig.  45. 

image  PZQZ  of  PQ  must  be  in  the  same  plane  as  the  microm- 
eter. If,  therefore,  a  finely  divided  scale  (usually  called  a 
stage  micrometer)  is  taken  for  the  object  PQ,  the  observer 
can  at  once  see  how  many  divisions  on  this  image  correspond 
with  one  division  on  the  micrometer,  and  so  ascertain  how 
long  an  object  must  be  in  order  that  its  real  image  may  be 
equal  to  the  distance  between  two  lines  on  the  micrometer. 
The  micrometer  has  sometimes  a  number  of  small  equal 
squares  ruled  on  it  in  place  of  a  scale,  and  by  means  of  these 
the  number  of  small  objects,  such  as  colonies  of  bacteria, 
occupying  a  given  area  can  be  ascertained.  The  magnifying 


64 


EXPERIMENTAL    PHYSICS. 


power  of  the  objective,  together  with  the  field  glass,  depends 
on  the  distance  between  them,  and  it  should  be  noted  that 
in  finding  the  length  of  a  small  object  by  the  method  just 
described,  this  distance  should  be  the  same  as  in  finding  the 
number  of  divisions  on  the  stage  micrometer  corresponding 
to  one  on  that  in  the  eyepiece. 

The  Telescope.  —  Expressions  can  also  be  calculated  for  the 
magnifying  power  of  telescopes.  In  the  case  of  the  astronomi- 
cal telescope  it  can  be  shewn  to  be  the  ratio  F:f,  where  F 
is  the  focal  length  of  the  objective,  and  /  that  of  the  eyepiece. 
Practically  the  magnifying  power  of  a  telescope  is  determined 
by  looking  through  it  with  both  eyes  open,  at  a  distant  scale, 
or  some  object  marked  with  equal  divisions,  such  as  a  picket 
fence.  The  number  of  divisions  on  the  scale  corresponding 
with  one  on  the  image  is  taken  as  a  measure  of  the  enlarge- 
ment. 


XXIV.     EXERCISES   WITH   PHOTOGRAPHIC   LENSES. 

If  the  light  from  objects  situated  outside,  such  as  trees  and 
buildings,  be  allowed  to  enter  a  darkened  room  through  a  small 
hole  in  a  shutter,  there  will  be  formed  on  a  white  screen  placed 
before  the  hole,  or  on  the  opposite  wall  of  a  room,  an  inverted 
./7  image  in  true  colours  of  the 

outside  objects.  This  phenom- 
enon is  due  to  the  rectilinear 
propagation  of  light,  and  can 
be  easily  explained. 

In  Fig.  46  AB  is  an  object 
situated  outside,  A'B'  is  its 
image  formed  on  a  screen,  and 
P  is  a  small  aperture  in  the 
shutter.  A  pencil  of  light  coming  from  the  point  A  will,  on  pass- 
ing through  the  opening,  form  a  spot  of  light  at  A'  which  cannot 
be  distinguished  from  a  single  point  if  P  is  extremely  small. 


Fig.  46. 


EXERCISES   WITH    PHOTOGRAPHIC    LENSES.  65 

As  no  other  point  on  AB  can  send  light  to  A',  its  colour  and 
brightness  will  therefore  depend  on  that  of  A,  and  it  may  thus 
be  said  to  be  the  image  of  A.  Similarly,  B'  will  be  the  image 
of  B,  and  all  points  between  A  and  B  will  have  their  images 
between  A'  and  B'.  Hence,  when  light  is  admitted  to  a  dark 
room  through  a  small  opening,  we  are  able  to  form  an  inverted 
image  of  the  outside  objects.  As  the  image  of  each  external 
point  is  in  reality  a  spot  and  not  a  point,  it  is  evident  that  if 
the  opening  of  P  is  large,  those  formed  by  the  light  coming 
from  different  points  on  the  outside  objects  will  overlap,  and 
hence  no  image  will  be  produced.  It  is  for  this  reason,  there- 
fore, that  no  image  is  obtained  when  the  light  is  allowed  to 
enter  a  room  through  a  large  opening,  such  as  a  window. 

The  phenomenon  just  described  affords  the  simplest  method 
of  taking  a  photograph.  If  the  lens  be  removed  from  a  camera, 
and  a  sheet  of  tin-foil  pierced  with  a  very  fine  hole  be  placed 
over  the  opening,  an  image  can  be  formed  on  a  sensitized  plate, 
and  a  negative  thus  obtained.  Although  no  "focussing"  is 
required  by  this  method,  yet  on  account  of  the  small  amount 
of  light  forming  the  image,  the  length  of  time  required  for  a 
proper  exposure  makes  it  generally  impracticable. 

The  Use  of  the  Lens.  —  When  a  lens  is  used,  it  is  because  it 
produces  an  image  of  greater  intensity,  and  the  times  of  expos- 
ure are  therefore  correspondingly  diminished.  Intensity,  how- 
ever, is  then  obtained  at  the  expense  of  distinctness,  and  the 
images  formed,  owing  to  chromatic  and  spherical  aberration,  are 
far  from  being  perfect.  To  remove  the  defects  so  produced,  or 
at  least  reduce  them  to  a  minimum,  opticians  have  had  recourse 
to  the  use  of  compound  lenses  and  of  stops. 

Chromatic  Aberration.  —  When  white  light  passes  from  one 
medium  into  another,  the  violet  rays  composing  it  undergo, 
owing  to  their  greater  refrangibility,  a  greater  deviation  than 
the  red  rays,  and  the  defect  caused  by  this  in  images  formed 
with  lenses  is  called  chromatic  aberration.  This  dispersion, 
or  separation  of  the  rays  of  different  colours,  varies  with  the  two 


66  EXPERIMENTAL   PHYSICS. 

media,  and  it  can  be  readily  shewn  by  experiments  with  prisms 
of  different  substances,  and  of  different  angles  that,  when  the 
dispersive  effects  of  two  prisms  are  equal,  their  refractive 
effects  are,  in  general,  not  so,  and  that  therefore  by  suitably 
combining  two  prisms  of  different  refracting  angles,  but  hav- 
ing equal  dispersive  effects,  we  may  produce  refraction  of  white 
light  without  causing  dispersion.  It  therefore  follows,  since  by 
taking  radial  sections  of  a  lens  we  may  shew  its  general  effect 
to  be  the  same  as  that  of  a  prism,  that  by  suitably  combining 
two  lenses  of  different  substances  we  can  bring  rays  of  light 
to  a  focus  without  the  presence  of  colour  effects.  Hence  the 


Fig.  47. 

method  of  constructing  achromatic  combinations  by  uniting 
lenses  made  from  different  kinds  of  glass. 

A  simple  and  instructive  exercise  in  this  connection  is  to  test 
the  objectives  of  microscopes  and  telescopes  for  achromatism 
by  allowing  the  light  from  a  strongly  illuminated  pin-hole  to 
pass  through  the  combination.  The  emerging  rays  may  then 
be  examined  by  receiving  them  through  a  simple  eyepiece,  or 
by  allowing  them  to  form  an  image  on  a  white  screen. 

Spherical  Aberration.  —  When  rays  of  light  issuing  from  an 
illuminated  point  fall  upon  a  convex  lens,  they  do  not,  on  being 
refracted,  all  intersect  in  a  single  point,  but,  as  Fig.  47  indicates, 
in  many  of  them,  situated  at  different  distances  from  the  lens ; 
and  each  of  these  may  be  taken  to  be  the  image  of  the  point 
from  which  the  rays  emanate.  This  is  well  exhibited  by  cover- 


EXERCISES   WITH    PHOTOGRAPHIC   LENSES.  67 

ing  the  lens  with  a  screen  in  which  are  cut  a  number  of  concen- 
tric rings.  If  the  light  be  allowed  to  pass  through  only  one 
ring  at  a  time,  it  will  be  found  that  in  order  to  focus  an  image 
the  screen  must  be  placed  nearer  to  the  lens,  according  as  the 
radius  of  the  ring  through  which  the  light  passes  increases. 

From  this  it  will  be  evident  that  when  a  convex  lens  is  used 
to  form  an  image  of  an  object,  there  is  in  reality  a  multiplicity 
of  images  formed  of  each  point;  these  images  being  situated 
close  to  each  other,  but  yet  in  different  planes  perpendicular  to 
the  axis  of  the  lens.  When,  then,  an  image  of  an  object  is 
focussed  on  a  screen,  it  is  in  fact  only  so  for  the  set  of  images 
which  happens  to  lie  in  the  plane  of  the  screen.  Some  of  the 
rays  are  focussed  in  front  of  it,  and  some  behind  it,  and  these 
on  striking  the  screen  produce  a  blurred  appearance,  and  the 
image  is  thus  lacking  in  what  is  technically  called  definition. 
This  defect  in  images  is  said  to  be  due  to  spherical  aberration, 
and,  in  order  to  minimize  its  effects,  recourse,  as  previously 
indicated,  is  had  to  the  use  of  stops.  It  will  be  quite  clear  that 
whether  these  are  placed  in  front  of  the  lens,  or  behind  it,  their 
effect  will  be  to  cut  off  some  of  the  rays  which  do  not  focus  on 
the  screen,  or  sensitized  plate,  and  in  this  way  produce  better 
definition.  If  it  were  desired  to  produce  an  image  free  entirely 
from  this  blurred  appearance,  it  would  be  necessary  to  use  a 
stop  with  a  very  small  opening;  and  although  the  image  then 
formed  would  be  sharp  and  distinct,  the  illumination  would  be 
weak,  and  there  would  be  little,  if  any,  advantage  over  the  ordi- 
nary pin-hole  method.  In  reducing  the  effects  of  spherical 
aberration  stops  are  used  which,  while  producing  images  not 
perfectly  distinct,  give  moderate  intensity,  and  so  permit 
short  exposures.  For  practical  purposes  the  intensities  of  the 
image  may  be  taken  proportional  to  the  areas  of  the  open- 
ings in  the  stops,  and  it  is  an  instructive  exercise  to  cal- 
culate the  relative  times  of  exposure  for  a  set  of  stops,  and  to 
test  these  by  taking  photographs  of  an  object  under  constant 
illumination. 


68  EXPERIMENTAL  PHYSICS. 

By  placing  the  screen  with  concentric  circular  openings  on 
different  lenses,  it  may  be  shewn  that  spherical  aberration 
depends  on  the  curvature  of  the  lens,  and  that  it  may  be 
diminished  by  using  two  lenses  of  small,  instead  of  one  of 
large,  curvature.  These  lenses  can  be  so  chosen  that  if  they 
are  placed  a  short  distance  apart  they  will  have,  in  combina- 
tion, a  focal  length  equivalent  to  that  of  a  single  lens ;  and 
since  by  this  arrangement  there  is  a  gain  in  definition  without 
a  counterbalancing  loss  in  intensity,  it  is  generally  adopted  in 
the  construction  of  photographic  objectives. 


XXV.    BUNSEN'S   PHOTOMETER. 

Although  we  are  not  able  to  measure  absolutely  the  intensity 
of  the  light  coming  from  a  given  source,  we  can,  however, 
compare  it  with  that  from  another.  The  eye,  which  cannot 
estimate  directly,  with  any  degree  of  accuracy,  the  relative 
intensity  of  two  sources  of  light,  is  a  good  judge  of  the  illumi- 
nations which  they  can  produce ;  and  it  is  on  the  principle  of 
equality  of  illuminations  that  the  photometers  ordinarily  em- 
ployed are  constructed. 

The  intensity  of  illumination  is  taken  proportional  to  light 
intensity,  and  is  defined  to  be  the  quantity  of  light  received  on 
a  unit  area  of  surface.  The  fundamental  law  in  photometry, 
which  is  that  of  the  inverse  square,  is  readily  demonstrated  by 
considering  a  small  cone  of  rays  coming  from  an  illuminated 
point.  Since  the  -same  amount  of  light  passes  through  any 
right  section  of  this  cone,  the  intensity  at  any  point  will  vary 
inversely  as  the  area  of  the  right  section  containing  the  point ; 
and  again,  as  any  such  section  varies  directly  as  the  square  of 
its  distance  from  the  vertex  of  the  cone,  the  law  is  evident. 

In  Bunsen's  photometer  the  light  from  some  standard, 
and  that  from  the  source  to  be  tested  are  allowed  to  fall 
perpendicularly  upon  the  opposite  sides  of  a  sheet  of  paper 
with  an  oiled  spot  on  it.  When  light  falls  upon  such  a 


BUNSEN'S   PHOTOMETER.  69 

paper,  more  of  it  is  reflected,  and  less  transmitted  by  the 
unoiled  portion  than  by  the  oiled,  and  hence  the  latter,  when 
viewed  from  the  side  next  the  source  of  light,  appears  darker 
than  the  remainder  ;  while  if  viewed  from  the  other  side,  it 
appears  brighter.  If  then  light  be  allowed  to  fall  on  the  two 
sides  of  the  paper  so  that  both  sides  are  equally  illuminated, 
that  from  one  source,  which  was  lost  by  transmission,  is  restored 
by  the  light  transmitted  from  the  other. 

When  the  lights  have  been  so  adjusted  that  this  condition 
obtains,  the  oiled  part  can  no  longer  be  distinguished  from 
the  rest  of  the  paper.  Let,  then,  d  be  the  distance  of  the 
standard  from  the  paper,  and  d1  that  of  the  light  to  be  tested  ; 
A  and  A  '  the  intensities  of  these  two  lights  respectively  at 
a  unit  distance,  and  B  and  B'  the  intensities  at  the  paper 
disc. 

By  the  law  of  the  inverse  square  : 

B       \  B'       i 


If,  therefore,  the  intensities  are  the  same  at  the  paper,  B=B', 
and  .-.A'=—2A. 

The  intensity  of  the  tested  light  is  then  —  times  that  of 
the  standard. 

In  making  a  test,  different  values  should  be  given  to  d,  and 
corresponding  ones  found  for  d',  the  mean  of  the  results  so 
obtained  being  taken  as  the  intensity  of  the  light  examined. 
As  it  is  essential  that  no  light  should  fall  upon  the  paper 
disc  except  that  coming  from  the  two  sources  mentioned,  the 
room  in  which  the  experiment  is  conducted  should  be  thor- 
oughly darkened,  and  its  walls  painted  black.  A  simple  device 
to  permit  both  sides  of  the  paper  disc  to  be  seen  simultaneously 


70  EXPERIMENTAL    PHYSICS. 

is  to  place  it  in  a  vertical  frame,  to  the  back  of  which  are 
attached  two  mirrors  inclined  at  an  angle  to  the  paper  such  that 
the  observer  can  by  looking  into  the  mirrors,  see  images  of  both 
sides  of  the  paper  at  the  same  time.  The  oiled  spot  is  gener- 
ally made  circular ;  but  it  is  well  to  check  results  by  making 
it  of  some  irregular  form.  A  modification  of  this  is  obtained 
by  placing  together  three  sheets  of  paper,  in  the  center  one 
of  which  is  a  hole  of  any  desired  shape.  Owing  to  the  pres- 
ence of  light  reflected  from  the  walls  of  the  room,  and  to 
unequal  absorption  by  the  two  parts  of  the  disc,  it  is  rarely 
possible  to  arrange  the  two  lights  so  that  the  oiled  spot  entirely 
disappears.  Indeed,  the  results  obtained  by  the  method  are 
only  approximate  at  best.  It  is  assumed  in  the  theory  that 
the  sources  of  light  are  points,  and  that  the  lights  are  of  the 
same  quality,  — conditions  which  do  not  hold  in  practice. 

Various  forms  of  standard  lights  have  been  devised  ;  but 
what  seems  to  give  the  steadiest  light  is  to  illuminate  a  small 
hole  in  a  screen  by  means  of  the  flame  of  a  coal-oil  lamp  burn- 
ing a  broad  wick.  As,  in  testing  gas  flames  and  incandescent 
lamps,  it  is  found  that  the  intensity  at  a  given  distance  varies 
with  the  position  of  the  flame  in  the  one  case,  and  with  that 
of  the  carbon  filament  in  the  other,  the  student  should  make  a 
table  of  his  results,  or  plot  a  curve  shewing  the  intensity  of  the 
light  in  different  positions.  Bunsen's  photometer  may  also  be 
used  to  compare  the  absorptive  powers  of  different  transparent 
media,  such  as  glass,  or  water,  by  finding  the  intensities  of  the 
light  transmitted  by  them  from  a  given  source.  The  results 
obtained,  however,  will  not  be  exact,  as  light  is  always  reflected 
from  the  surface  of  the  medium  employed. 

XXVI.     AYRTON'S   PHOTOMETER  —  MODIFIED  FROM  BUNSEN'S. 

When  the  light  to  be  examined  is  very  great  compared  with 
that  of  the  standard,  the  experiment  takes  up  so  much  room,  if 
the  previous  method  is  adopted,  that  it  is  quite  impracticable. 


AYRTON'S    PHOTOMETER.  71 

This  difficulty  has  been  overcome  by  Ayrton,  who  introduces 
a  biconcave  lens  between  the  light  tested  and  the  photometer. 

Theory.  —  In  Fig.  48  P  is  the  source  of  the  light  to  be 
tested,  AB  the  lens,  ST  the  paper  disc  of  the  photometer, 
and  CPD  a  cone  of  light  striking  the  lens.  Let  A  and  A'  be 
the  intensities  of  the  standard  and  of  the  given  light  at  a  unit 
distance  ;  B  and  B'  the  intensities  of  the  light  from  the  latter 
at  the  disc,  with  the  lens  interposed,  and  with  it  removed 
respectively,  and  B"  that  of  the  standard  at  the  disc.  Also  let 
C  be  the  area  of  the  circle  of  light  on  the  disc  formed  by  the 
cone  of  rays  DPC  with  the  lens  inserted,  and  C'  with  it  removed; 

s 


Fig.  48. 

and  let  a  be  the  distance  from  the  paper  disc  to  the  light  tested, 
c  that  to  the  standard,  and  /  that  to  the  lens. 

Since  the  intensity  of  illumination  is  the  amount  of  light  on 
a  unit  area,  and  since  the  total  amount  of  light  on  C  and  C', 
under  the  conditions  mentioned  above,  may  be  taken  to  be  the 
same,  we  have 

BC=B'Ct  or  B'=~B. 

(^ 

Again,  from  the  fundamental  law  we  have 

—  =02,  or  A'=a2B'; 
B' 


72  EXPERIMENTAL   PHYSICS. 

Again,  we  have      —  =  -L   Or  B"  =  ™ 

Therefore  when  the  two  sides  of  the  disc  are  equally  illu- 
minated, we  have  B'  =B, 

A^_A     .      A^_azC 
a*C~  c*  "'   A~*C 

Now  denote  the  circular  area  of  the  lens  through  which  the 
light  passes  by  C",  and  let  the  light  after  refraction  appear  to 
come  from  a  point  P'  at  a  distance  x  behind  the  lens. 

Then  =  71^'  (0 


Also,  since  P'  is  the  image  of  P, 


(3) 
j-t-a  —  t 

From  (i),  (2),  and  (3), 


i.e.  the  intensity  of  the  given  light  will  be 


times  the  standard. 

Experiment.  —  Place  the  biconcave  lens  in  a  screen,  and 
insert  it  between  the  light  to  be  tested  and  the  paper  disc 
of  the  photometer.  Then,  as  in  the  previous  method,  adjust 
the  standard  so  that  the  oiled  spot  cannot  be  distinguished 
from  the  rest  of  the  paper,  and  measure  the  lengths  a,  c,  and  /. 


ROTATING   DISC  PHOTOMETER. 


73 


By  substituting  in  the  formula  these  values  together  with 
that  of  f,  the  focal  length  of  the  lens,  which  may  be  found 
by  either  of  the  methods  given  in  Experiment  XXL,  the  in- 
tensity of  the  light  may  be  determined.  In  making  a  test, 
the  student  should  obtain  a  number  of  results  by  giving  dif- 
ferent values  to  a  and  /,  and  therefore  to  c. 

Ayrton's  method  is  especially  applicable  in  finding  the  in- 
tensity of  the  light  from  an  incandescent  lamp,  or  that  from  an 
electric  arc.  The  intensity  of  the  latter  should  be  examined 
with  the  carbons  making  different  angles  with  the  vertical, 
because,  just  as  in  the  case  of  gas  flames  and  incandescent 
lamps,  the  intensity  at  a  given  distance  varies  with  the  position 
of  the  arc. 


XXVII.     ROTATING   DISC  PHOTOMETER. 

When  a  number  of  objects  are  passed  in  rapid  succession 
before  the  eye  of  an  observer,  the  effect,  owing  to  the  persist- 


Fig.  49. 


ence  of  luminous  sensations,  is  the  same  as  if  the  retina  received 
the  impressions  from  the  different  objects  simultaneously;  and 
it  is  for  this  reason  that  we  are  able  to  produce  a  white  appear- 


74  EXPERIMENTAL  PHYSICS. 

ance  by  the  rapid  rotation  of  a  disc  of  coloured  sectors  arranged 
in  proper  proportions,  or  by  rapidly  oscillating  the  spectrum 
.produced  by  allowing  a  pencil  of  the  sun's  rays  to  fall  on  a 
prism.  The  principle  here  involved  has  been  applied  with  con- 
siderable success  to  photometric  investigations.  In  Fig.  49,  A, 
a  black  circular  disc  made  of  thin  sheet  metal,  or  of  cardboard, 
has  two  apertures  in  it,  whose  sides  are  circular  arcs  concentric 
with  the  disc  ;  B,  a  double  sector  of  the  same  material  as  the 
disc,  is  used  to  reduce,  or  enlarge  the  size  of  these  apertures ; 
and  C,  D,  two  double  sectors  of  some  gray-coloured  material,  are 
of  such  a  size  that  when  they  are  placed  concentric  with  the  disc 
A  the  lighter  one  C  extends  just  beyond  the  outer  edges  of  the 
apertures,  while  the  darker  one  D  extends  half-way  across  them. 
The  fundamental  hypothesis  in  the  work  with  rotating  discs 
is  that  the  amount  of  light  coming  from  a  sector  is  proportional 
to  the  angle  of  this  sector.  The  edges  of  the  plates  used  play 
a  considerable  part  in  these  investigations,  and  experiment 
seems  to  show  that,  if  the  angles  are  not  small,  and  the  edges 
of  the  plates  are  made  thin  by  bevelling,  the  assumption  is 
warrantable. 

EXERCISE  I.  —  Comparison  of  gray  tints. 

It  is  often  desirable  to  have  an  accurate  notion  of  the  relative 
quantities  of  light  reflected  from  different  papers  commonly 
called  white.  In  order  to  investigate  this,  cut  two  sectors  such 
as  C  and  D  from  the  two  sheets  of  paper  to  be  examined,  and 
place  them  on  the  same  axis  of  rotation  as  the  black  disc  A,  and 
behind  it,  the  darker  sector  being  next  to  the  disc.  If,  then, 
this  arrangement  be  rotated  in  front  of  a  black  screen  of  velvet, 
or  other  such  material,  there  will  appear  two  concentric  rings 
of  a  drab  colour  produced  by  the  combination  of  the  white  of  the 
sector  with  the  black  of  the  disc.  As  their  colour  depends  only 
on  the  amount  of  the  white  in  combination  with  the  black,  the 
two  rings  may  be  made  of  the  same  tint  by  taking  suitable  angles 
for  the  sectors  of  paper.  If  great  accuracy  is  desired,  the  aux- 


ROTATING  DISC   PHOTOMETER.  75 

iliary  sector  B  may  be  used.  If,  for  example,  it  is  found  that 
the  black  of  the  disc  combined  with  81°  of  the  white  of  one  sec- 
tor gives  nearly  the  same  tint  as  when  combined  with  80°  of 
that  of  the  other,  then,  by  adding  i°  to  each  of  the  sectors,  or 
by  taking  i°  off,  we  may  get  the  ratios  82  : 81  or  80 : 79,  one  of 
which  will  be  a  closer  approximation  to  the  exact  result.  If  a 
and  b  are  the  angles  of  the  two  white  sectors,  when  the  rings 
are  of  the  same  tint,  and  Q,  Q'  are  the  quantities  of  light  re- 
flected from  a  unit  angle  of  each  of  these  sectors,  then  aQ  =  bQ', 
or  the  ratio  b :  a  is  a  measure  of  the  relative  whiteness  of  the 
two  papers.  If  the  edges  of  the  apertures  are  divided  into 
degrees  and  parts  of  a  degree,  the  angles  a  and  b  may  be  read 
directly.  This  same  method  may  be  applied  to  a  study  of  the 
light  reflected  from  walls  of  different  gray  tints.  It  is  done  by 
spreading  thin  layers  of  the  plasters  used  on  different  sectors, 
and  then,  when  it  is  dry,  proceeding  just  as  in  the  paper  tests. 
In  conducting  these  experiments  it  can  be  seen  that  the  colour 
of  the  two  rings  alters  slightly  with  the  direction  of  the  inci- 
dent light,  and  with  the  position  of  the  observer.  It  will  be 
found  that  the  best  results  are  obtained  when  the  disc  is 
illuminated  with  rays  of  light  incident  parallel  to  the  axis  of 
rotation,  and  when  a  small  part  only  of  the  rings  is  viewed  by 
means  of  a  telescope. 

EXERCISE  II.  —  To  compare  the  intensities  of  light  coming 
from  two  independent  sources. 

To  perform  this  experiment  the  disc  should  have  two  concen- 
tric sets  of  apertures  a  short  distance  apart,  and  the  light  should 
be  allowed  to  fall  on  the  back  of  the  disc  in  such  a  manner  that 
each  of  the  sets  is  constantly  illuminated  by  the  light  from  one 
of  the  two  sources.  By  altering  the  size  of  these  apertures 
until  the  colours  of  the  rings,  formed  by  the  rotation  of  the  disc, 
are  the  same,  a  measure  of  the  intensities  of  the  light  from  the 
two  sources  may  be  obtained.  If  a  and  b  are  the  angular  open- 
ings, the  intensities  are  then  in  the  ratio  b :  a. 


76  EXPERIMENTAL   PHYSICS. 

As  considerable  difficulty  is  likely  to  be  met  with  in  arrang- 
ing suitable  mechanism  for  this  method,  a  better  one  is  to  use 
two  small  discs  with  a  single  set  of  apertures  in  each,  and  to 
rotate  them  in  the  same  plane.  In  this  way  the  illuminations 
may  be  made  more  simply,  and  more  nearly  under  the  same 
conditions.  It  is  evident  that  these  methods  only  apply  when 
the  lights  examined  are  of  the  same  quality.  Whenever  the 
question  of  colour  comes  in,  the  investigation  becomes  compli- 
cated and  uncertain. 

EXERCISE  III.  —  To  compare  the  absorptive  powers  of  differ- 
ent media. 

If  a  screen  be  made  partly  of  one  translucent  substance,  and 
partly  of  another,  and  light  from  a  single  source  be  allowed  to 
fall  on  the  back  of  this  screen,  then  that  which  comes  through 
may  be  considered  as  coming  from  two  independent  sources. 
The  intensity  of  the  light  coming  from  each  part  of  the  screen 
may  then  be  investigated  by  either  of  the  methods  outlined  in 
Exercise  II.,  and  the  absorptive  powers  of  the  different  sub- 
stances composing  the  screen  compared. 

In  Exercises  II.  and  III.  every  precaution  should  be  taken  to 
insure  that  no  light  falls  on  the  apertures  of  the  discs  except 
that  which  is  being  examined. 


XXVIII.     SPECIFIC   HEAT   OF   SOLIDS. 

The  Method  of  Mixture.  —  In  Fig.  50  B  is  a  boiler,  in  which 
water  may  be  heated,  and  AC  is  a.  steam  jacket  connected  to  it 
by  a  tube  indicated  in  the  figure.  To  the  upper  portion  of  the 
tube  DB,  leading  from  the  boiler  to  the  outside  air,  there  is 
fitted  a  water  jacket  through  which  cold  water  is  kept  constantly 
running.  The  calorimeter  G  consists  of  two  metallic  vessels, 
generally  made  of  sheet  brass,  the  one  resting  on  wooden  sup- 
ports inside  the  other.  The  solid  E,  whose  specific  heat  is 
required,  is  suspended  by  a  string  in  the  inclosure  F  within  the 


SPECIFIC   HEAT   OF   SOLIDS. 


77 


steam  jacket.  The  steam  after  leaving  the  boiler  passes  into 
this  jacket,  and  then  out  into  the  tube  DB,  where,  on  coming 
into  contact  with  the  cold  air,  it  condenses  and  falls  back  into 
the  boiler.  In  this  way  a  steady  supply  of  steam  is  led  into  the 
jacket,  whose  temperature  is  thus  kept  constant,  and  there  is, 
therefore,  no  occasion  for  renewing  the  water  in  the  boiler.  The 
steam  jacket  is  generally  provided  with  attachments  by  means 


Fig.  50. 

of  which  both  ends  of  the  inclosure  F  may  be  stopped,  and 
radiation  thus  prevented. 

The  unit  of  heat  generally  adopted  is  the  quantity  required 
to  raise  the  temperature  of  one  gram  of  water  one  degree  centi- 
grade, and  is  called  a  calorie.  It  is  found  that,  if  equal  masses  of 
two  different  substances  are  subjected  to  the  same  heat,  under 
the  same  circumstances,  for  a  given  time,  their  temperatures 
will  vary  considerably. 


78  EXPERIMENTAL   PHYSICS. 

This  shews  that  different  quantities  of  heat  must  be  imparted 
to  equal  masses  of  different  substances  to  make  the  same  altera- 
tion in  their  temperatures,  and  the  specific  heat  of  a  substance 
is  defined  to  be  the  ratio  of  the  quantity  of  heat  required  to  raise 
the  temperature  of  a  given  mass  of  this  substance  one  degree, 
to  that  required  to  raise  the  temperature  of  an  equal  mass  of 
water  one  degree. 

Theory Let  M  be  the  mass  of  the  solid  in  grams,  T°  its 

temperature  before  it  is  dropped  into  the  water;  m  and  m* 
the  masses  of  the  water  in  the  calorimeter,  and  of  the  calori- 
meter respectively;  f  and  6°  their  initial  and  final  tem- 
peratures, and  c,  and  c'  the  specific  heats,  respectively,  of 
the  given  solid,  and  of  the  substance  of  which  the  calorimeter 
is  made. 

The  heat  given  up  by  the  solid  will  then  be  equal  to 
cM(T—6]  calories,  while  that  gained  by  the  water  will  be  equal 
to  m(6  —  t}  calories,  and  that  by  the  calorimeter  m'c'(6—t}. 
Since  the  heat  lost  by  the  solid  is  gained  by  the  calorimeter 
and  the  water  it  contains,  we  have 

cM(  T-  (9)  =m(0-t)+  m  'c'  (0  -  /), 


or 


M(T-6) 


Water  Equivalent  of  the  Calorimeter.  —  The  expression  m'c' 
in  this  formula  is  called  the  water  equivalent  of  the  calorimeter, 
since  from  its  position  it  is  equivalent  to  an  addition  to  the 
mass  of  water  in  the  calorimeter.  Its  value  is  best  obtained  by 
finding  its  mass  m'  by  weighing,  and  by  taking  the  specific  heat 
c'  of  the  substance  of  which  it  is  made,  from  the  tables,  but  it 
may  also  be  found  experimentally  in  the  following  manner : 
Into  a  mass  M^  of  water  contained  in  the  calorimeter  (the 
temperature  of  both  being  t^}  pour  a  mass  of  water  M%  of 
temperature  /a°,  higher  than  t£,  and  let  6±  be  the  result- 


SPECIFIC   HEAT   OF   SOLIDS.  79 

ing  temperature.     By  the  same  process  of  reasoning  as  above 
we  will  have 


MM*  - 
m'c'=     2V  2 


i.e.    m'c'  is    expressed    in    terms   of   quantities    which   can   be 
directly  found. 

Experiment.  —  Weigh  the  solid  E  whose  specific  heat  is  to  be 
determined,  and  then  suspend  it  together  with  a  thermometer 
in  the  inclosure  within  the  steam  jacket,  as  indicated  in  the 
figure.  Care  should  be  taken  to  keep  a  constant  stream  of  cold 
water  running  through  the  vessel  attached  to  the  upper  part  of 
the  tube  DB,  and  not  to  fill  the  boiler  with  water  higher  than 
the  opening  in  the  tube  leading  from  it  to  the  steam  jacket. 
While  the  solid  is  being  heated  to  the  temperature  of  the 
steam,  the  student  may  find  the  water  equivalent  of  the 
calorimeter  experimentally.  In  conducting  this  preliminary 
part  of  the  experiment  care  should  be  taken  to  have  the  tem- 
peratures, and  the  quantities  of  water  involved  as  nearly  as 
possible  the  same  as  those  in  the  experiment  proper.  Weigh 
the  calorimeter,  including  thermometer  and  stirrer.  Partly  fill 
it  with  water  of  temperature  tf,  below  that  of  the  room,  and 
again  weigh  it.  The  difference  between  these  two  weights  will 
give  the  mass  of  water  Ml  in  it.  Then  pour  into  the  water  Ml 
a  sufficient  quantity  of  water  of  temperature  /2°  to  bring  the 
resultant  temperature  as  much  above  that  of  the  room  as  t±  was 
below  it.  In  this  way  the  loss  of  heat  due  to  radiation  may  be 
partly  overcome.  Again  weigh  the  calorimeter,  and  the  differ-: 
ence  between  this  weight  and  the  last  will  give  the  mass  M^  of 
the  water  poured  in.  The  quantities  t^,  /2,  0lt  Mlt  M^  thus 
being  known,  the  water  equivalent  may  be  determined  as  pre- 
viously indicated.  The  vessel  from  which  the  water  Mz  is 
poured  should  contain  more  than  is  actually  necessary,  and  t^ 


80  EXPERIMENTAL   PHYSICS. 

should  be  the  mean  of  the  temperatures  of  this  water  before 
and  after  Mz  is  poured  out.  Especial  care  should  be  taken  to 
pour  the  water  into  the  calorimeter  in  such  a  manner  that  the 
least  possible  amount  of  heat  is  lost  in  the  pouring. 

Having  determined  the  water  equivalent,  again  take  the 
calorimeter,  and  partly  fill  it  with  water  of  temperature  f  and 
weigh  it.  The  difference  between  this  weight  and  that  of  the 
calorimeter  empty  will  give  the  mass  m  of  the  water  poured 
into  it.  Having  made  certain  that  the  solid  is  now  at  the  tem- 
perature of  the  steam  jacket  T°,  which  is  noted  by  reading  the 
thermometer  suspended  within  the  inclosure  F,  or  by  finding 
from  tables  the  boiling-point  of  water  corresponding  to  the 
reading  of  a  barometer  suspended  in  the  room,  slide  the  calo- 
rimeter under  the  steam  jacket.  Then  drop  the  solid  into  it, 
and  stir  the  water  gently.  The  temperature  of  the  water  will 
rise  gradually,  then  remain  stationary  for  a  time,  and  finally 
begin  to  lower.  The  stationary  temperature  0°  will  then  be  that 
resulting  from  plunging  the  heated  solid  into  the  water.  The 
mass  M  of  the  solid  having  been  previously  found,  and  m,  m'c', 
6,  t,  and  T  thus  being  known,  we  can,  by  applying  the  formula, 
obtain  the  specific  heat  of  the  solid. 

XXIX.     SPECIFIC   HEAT   OF   LIQUIDS. 

In  Fig.  51,  AB  is  a  boiler  containing  water,  and  CD  is  a 
hollow,  metallic  reservoir  almost  wholly  immersed  in  it.  The 
calorimeter  consists  of  three  parts,  EF  resting  on  a  stand,  GH 
resting  on  wooden  supports  inside  of  EF,  and  K  immersed  in 
water  contained  in  GH.  The  different  parts  of  the  calorimeter 
are  generally  made  of  thin  brass,  the  part  K  being  made  of  such 
a  form  as  to  expose  as  much  surface  as  possible  to  the  water. 
As  the  figure  indicates,  it  is  connected  to  the  reservoir  by  a 
tube  in  which  there  is  a  tap. 

Theory.  —  In  this  case,  also,  the  method  is  one  of  mixture. 
The  liquid  whose  specific  heat  is  to  be  determined,  is  first  heated 


SPECIFIC   HEAT   OF   LIQUIDS. 


81 


in  the  reservoir  CD,  and  then  allowed  to  run  into  the  calo- 
rimeter, where  it  gives  up  its  heat  to  the  water,  and  the  vessel 
containing  it.  Let  Mv  M2,  and  M  be  the  masses,  in  grams,  of 
the  calorimeter,  the  water  in  it,  and  the  given  liquid  respec- 
tively, /j0  and  6°  the  initial  and  final  temperatures  of  the  water 
in  the  calorimeter,  and  t°  the  temperature  of  the  liquid  before 
it  is  run  into  the  calorimeter.  Also  let  c  and  cl  be  the  specific 


Fig.  51. 

heats  of  the  given  liquid  and  of  the  metal  of  which  the  calo- 
rimeter is  made,  respectively. 

The  heat  given  out  by  the  liquid,  on  being  run  into  the 
calorimeter,  is  then  equal  to  cM(t  —  6}  calories,  while  that 
gained  by  the  calorimeter  is  clM1(0  —  ^l),  and  that  by  the 
water  in  it  M0  —  f. 


or 


M 


(t-6} 


82  EXPERIMENTAL  PHYSICS. 

Water  Equivalent  of  the  Calorimeter.  —  In  this  experiment, 
as  in  the  last,  clMl  is  the  water  equivalent  of  the  calorimeter, 
and  may  be  found  by  either  of  the  methods  there  given.  If 
the  experimental  method  is  adopted,  the  error  in  this  case  aris- 
ing from  the  water  cooling  when  being  poured  into  the  calo- 
rimeter, may  be  avoided  by  heating  it  in  the  reservoir  CD, 
and  then  allowing  it  to  run  through  the  tube  connecting  the 
two  vessels. 

Experiment.  —  First  weigh  the  two  parts  of  the  calorimeter 
GH  and  K,  together  with  the  stirrer  and  a  thermometer. 
Then  pour  in  water  sufficient  to  cover  the  part  K,  as  indicated 
in  the  figure,  and  again  weigh  them.  The  difference  between 
the  two  weights  will  give  the  mass,  Mv  of  the  water  poured  in. 
In  the  mean  time,  suspend  a  thermometer  in  the  boiler,  and 
place  in  the  reservoir  CD  enough  of  the  liquid  whose  specific 
heat  is  to  be  determined  to  nearly  fill  the  vessel  K,  and  allow 
it  to  become  heated  by  the  surrounding  water. 

In  order  to  prevent  evaporation,  care  should  be  taken  to  keep 
the  top  of  the  reservoir  closed  during  the  heating  process,  and 
not  to  let  its  temperature  rise  above  the  boiling-point  of  the 
given  liquid  corresponding  to  atmospheric  pressure.  When  it 
has  been  sufficiently  heated,  the  temperature  t°  of  this  liquid 
may  be  noted  by  reading  the  thermometer  suspended  in  the 
water.  Also  note  the  temperature  tf  of  the  water  in  the  calo- 
rimeter, and  then  turn  the  tap,  and  allow  the  liquid  to  run  into 
it  through  the  tube. 

Gently  stir  the  water  in  the  calorimeter,  and  when  its  tem- 
perature has  become  stationary,  note  its  value  6°.  Then  weigh 
the  calorimeter  and  contents,  and  the  difference  between  this 
weight  and  the  last  will  give  the  mass  M  of  the  liquid  that  ran 
in.  This  result  may  be  checked  by  first  weighing  the  liquid 
before  it  is  poured  into  the  reservoir.  The  student  will  thus 
see  that  the  only  essential  difference  between  this  experiment, 
and  the  last  is  that  the  substance  whose  specific  heat  is  desired, 


LATENT   HEAT   OF   FUSION   OF   ICE.  83 

is  not  in  this  case  allowed  to  come  in  contact  with  the  water  in 
the  calorimeter. 

Especial  care  should  be  taken  to  choose  conditions  favourable 
to  overcoming  the  errors  due  to  evaporation,  as  in  this  experi- 
ment considerable  inaccuracies  of  this  kind  are  almost  certain 
to  be  met  with  by  the  student. 

The  specific  heat  of  a  given  liquid  may  also  be  calculated  by 
finding  the  specific  heat  of  a  solid  relative  to  it,  and  also  relative 
to  water.  If  Cl  and  £72  are  these  two  specific  heats  respectively, 
and  C  the  specific  heat  of  the  liquid,  we  have 


or 


XXX.     LATENT   HEAT   OF   FUSION   OF    ICE. 

If  a  certain  mass  of  water  at  o°  C.  is  placed  in  one  beaker,  and 
an  equal  mass  of  ice  at  o°  C.  in  another,  and  if  the  two  are  then 
placed  in  a  hot-water  bath,  it  is  found  that  at  the  instant  the  ice 
is  all  melted  the  temperature  of  the  water  produced  by  it  is 
still  o°  C.,  while  that  of  the  water  in  the  other  beaker  is  about 
80°  C.  From  the  similarity  of  the  conditions  to  which  the  two 
vessels  are  subjected,  it  is  evident  that  the  same  quantity  of 
heat  passes  into  each  during  the  time  they  are  in  the  hot  water. 
In  the  one  case  the  heat  is  used  up  in  raising  the  temperature 
of  a  substance,  while  in  the  other  it  is  used  up  in  changing  the 
physical  structure  of  the  substance  without  altering  its  tempera- 
ture. From  this  fact  the  heat  absorbed  in  the  latter  case  has 
been  called  latent  heat,  and  technically  the  latent  heat  of  fusion 
of  a  substance  is  defined  to  be  the  number  of  units  of  heat 
required  to  convert  a  unit  mass  of  it  from  the  solid  to  the 
liquid  state  without  raising  its  temperature.  Adopting  the 
units  of  heat  indicated  in  Experiment  XXVIII.,  the  latent 
heat  of  fusion  of  ice  is  the  number  of  calories  required  to  con- 


84  EXPERIMENTAL   PHYSICS. 

vert  one  gram  of  ice  at  o°  C.  into  water  without  altering  its 
temperature. 

Theory.  —  Into  a  quantity  of  water  M  contained  in  a  calo- 
rimeter of  mass  Mlt  drop  a  mass  of  ice  M2  of  temperature  o°  C. 
Let  /j0  and  6°  be  the  initial  and  final  temperatures  of  the  water 
respectively,  L  the  latent  heat  of  fusion  of  ice,  and  ^  the  specific 
heat  of  the  substance  of  which  the  calorimeter  is  made.  LMZ 
calories  is  then  the  quantity  of  heat  absorbed  in  melting  the 
ice,  and  MZ6  calories  that  in  raising  the  temperature  of  the 
water  produced  by  it  from  o°  C.  to  6°  C.  Since  the  quantity 
of  heat  given  up  by  the  calorimeter  and  the  water  in  it  is 
-  (9),  we  have 


-0)=LM2 
L=(M 


Experiment.  —  The  masses  M,  Mly  and  M%  are  found  by  weigh- 
ing, and  just  as  in  the  two  previous  experiments,  the  water 
equivalent  of  the  calorimeter  c-^M^  may  be  determined  either 
by  a  reference  to  a  table  of  specific  heats,  or  by  the  experi- 
mental method.  The  initial  temperature  t^,  and  the  masses  M 
and  M2  of  the  water  and  ice  respectively,  should  be  so  selected 
that  the  final  temperature  6°  is  as  much  below  the  tempera- 
ture of  the  room  as  t-£  is  above  it.  These  quantities  may 
be  ascertained  approximately  by  a  calculation,  but  the  student 
will  find  it  exceedingly  instructive  to  adopt  a  tentative  proc- 
ess, and  in  this  way  arrive  at  the  proper  proportions  that 
should  obtain  in  an  exact  determination  of  L.  As  it  is  essen- 
tial that  the  ice  should  melt  as  rapidly  as  possible,  it  is  best  to 
break  it  up  into  small  pieces,  and  to  dry  each  piece  thoroughly 
with  blotting  paper  before  it  is  placed  in  the  calorimeter.  Care 
should  also  be  taken  to  have  the  temperature  of  the  ice  at 
o°  C.,  by  placing  it  in  a  warm  room  a  short  time  before  the 
experiment  is  commenced. 


LEVEL   TESTING.  85 


XXXI.    LEVEL   TESTING. 

The  spirit  level  is  made  of  a  glass  tube  slightly  curved,  so 
that  its  axis  forms  part  of  a  circle  of  large  radius.  Usually 
when  tubes  come  from  the  manufacturer  they  have  a  slight 
curvature  at  each  point,  and  it  is  only  necessary  to  perfect  this 
in  the  piece  chosen  by  grinding  the  inside  with  emery  powder. 
The  tube  is  filled  with  alcohol,  or  ether,  excepting  a  small  space 
containing  a  bubble  of  air  which  tends  to  occupy  the  highest 
part  of  the  tube.  It  is  sealed  as  shewn  in  Fig.  52,  and  then 
firmly  cemented  in  a  brass  tube,  which  is  itself  attached  to  a 
brass  plate  (Fig.  53)  in  such  a  manner  that  when  the  plate 
rests  on  a  level  surface  the  air  bubble  remains  stationary  at  the 
center  of  the  opening  in  the  upper  side  of  the  brass  tube.  In 


Fig.  52.  Fig.  53. 

some  levels  a  thin  brass  strip  across  the  opening  marks  this 
position  of  the  bubble ;  while  in  others  its  position  is  marked 
by  the  central  division  of  a  scale  ruled  on  the  glass. 

For  fine  work,  a  level  should  be  very  sensitive,  and  should  be 
ground  to  a  true  curvature,  which  is  indicated  by  a  uniform 
run  of  the  bubble  when  it  is  given  a  slight  inclination  to  the 
horizon.  As  the  sensitiveness  of  a  level  should  be  in  strict 
keeping  with  the  instrument  to  which  it  is  attached,  it  is  neces- 
sary that  each  one  should  be  thoroughly  tested  before  being 
used  for  any  particular  work. 

Buff  and  Berger  s  Level  Trier,  an  instrument  for  this  pur- 
pose, is  shewn  in  Fig.  54.  It  consists  of  an  iron  plate  upon 
which  is  mounted  an  iron  bar  having  at  one  end  two  pivotal 
centers  resting  in  receptacles  provided  for  them  in  the  base 
plate,  and  at  the  other  a  micrometer  screw  carrying  a  graduated 
disc.  The  bar  is  provided  with  fixed  wyes  in  which  levels  to 


86 


EXPERIMENTAL  PHYSICS. 


be  tested  may  be  placed,  and  also  with  a  scale,  to  be  used  in 
case  none  is  ruled  on  the  level.     The  points  of  the  pivots  and 

of  the  micrometer  screw  are 
made  very  hard,  and  the  latter 
bears  on  a  plate  with  a  hard 
and  polished  surface.  This 
plate  is  often  made  to  move 
eccentrically  with  regard  to  the 
screw,  so  that  the  point  of  rest 
can  be  changed  in  case  of  wear. 

Theory.  — Let  x  be  the  height 
through  which  the  left-hand  side 
of  the  instrument  is  raised  on 
turning  the  micrometer  screw, 
and  let  y  be  the  distance  be- 
tween the  screw  point  and  a 
point  midway  between  the  two 
pivots.  The  circular  measure 
of  the  angle  through  which  the 
instrument  has  been  turned  is 

then  — ,  and  if  this  ratio  is  multi- 

y 

plied  by  3437.748  or  206264.9, 
the  angle  is  given  in  minutes  or 
seconds  respectively. 

EXERCISE  I.  —  To  investigate 
the  run  of  the  bubble. 

Place  the  level  to  be  tested  on 
the  wyes  on  the  bar,  and  then 
raise,  or  lower  the  latter  until 
the  bubble  is  at  one  end  of  the 
scale,  or,  if  the  level  has  no 
scale,  at  the  point  which  is  the 


LEVEL  TESTING. 


limit  of  the  run  of  the  bubble  in  practice.  The  micrometer 
disc  is  then  turned  over  equal  spaces,  and  the  run  of  the  bubble 
carefully  noted.  When  the  bubble  has  been  moved  over  its 
course,  it  should  be  moved  in  the  opposite  direction  in  the  same 
manner,  and  the  whole  operation  repeated  several  times.  The 
mean  value  of  all  the  observations  may  then  be  determined,  and 
the  value  of  one  division  on  the  level  expressed  in  minutes  or 
seconds  of  an  arc,  as  the  case  may  be.  If  the  level  to  be  tested 
cannot  be  easily  removed,  the  entire  instrument  to  which  it  is 
attached  may  be  placed  on  the  trier,  resting  on  points  provided 
for  it  directly  over  the  pivots. 

In  the  following  table,  exhibiting  the  manner  in  which  a  test 
was  made,  A  and  B  are  the  right  and  left  hand  sides  of  the  air 
bubble  respectively : 


Number  of 
Trials. 

Microm- 
eter 
Readings. 

Level  Scale  Readings. 

Differences. 

Length  of 
Bubble. 

A  End. 

BEnd. 

A  End. 

BEnd. 

I 

7 

9.8 

51.8 

4-4 

4.6 

6l.6 

2 

17 

14.2 

47.2 

4-3 

4-5 

61.4 

3 

27 

18.5 

42.7 

4-  S 

4.2 

6l.2 

4 

37 

23.0 

.38.5 

4.0 

4.1 

6l.S 

5 
6 

7 

47 
57 
67 

27.0 
31-5 

35-8 

34-4 
30.0 

25-7 

4-5 
4-3 
4-4 

4.4 
4-3 
4-7 

61.4 
6l.5 
6l.5 

8 

77 

40.2 

21.0 

6l.2 

8 

77 

40.0 

21.2 

61.2 

7 

67 

35-5 

25-5 

4-5 

4-3 

61.0 

6 

57 

30.2 

4.4 

4-7 

61.3 

5 

47 

26.7 

34-6 

4-4 

4-4 

61.3 

4 

37 

22.3 

38.8 

4-4 

4.2 

61.1 

3 

27 

18.2 

42.8 

4.1 

4.0 

61.0 

2 

*7 

14.0 

47.1 

4.2 

4-3 

61.1 

I 

7 

9-5 

Sl-S 

4-5 

4-4 

61.0 

60.9 

61.1 

Mean  value  of  differences,  4.36  twentieths  of  an  inch. 

The  pitch  of  the  micrometer  screw  in  the  instrument  used 
was  one-sixtieth  of  an  inch,  its  disc  divided  into  one  hundred 


88  EXPERIMENTAL   PHYSICS. 

parts,  the  level  scale  graduated  to  twentieths  of  an  inch,  reading 
right  and  left  from  zero  at  the  center,  and  the  distance  between 
the  micrometer-screw  point  and  the  line  joining  the  two  pivots 
was  17.9  inches.  By  a  simple  calculation  it  may  be  seen  that 
the  angle  corresponding  to  a  rotation  of  10  divisions  on  the 
micrometer  disc  is  19.2  seconds,  and  that  therefore  an  inclina- 
tion of  4.4  seconds  caused  the  bubble  in  the  level  tested  to  move 
over  one-twentieth  of  an  inch.  This  then  may  be  taken  as  an 
indication  of  the  sensitiveness  of  the  level.  The  uniformity 
exhibited  in  the  displacements  of  the  bubble  shews  that  the 
curvature  of  the  level  was  very  regular. 

EXERCISE  II.  —  To  find  the  radius  of  curvature  of  a  level. 

This  is  found  by  rotating  the  level  through  some   known 
angle,  and  noting  the  displacement  of  the  bubble  produced.     If 

z  is  this  displacement,  and  -  the  circular  measure  of  the  angle 
of  rotation,  the  radius  of  curvature  of  the  level  is  given  by 


A  great  source  of  error  in  spirit  levels,  increasing  with  their 
sensitiveness,  is  an  unequal  heating  of  the  level  tube.  The 
bubble  will  always  move  towards  the  warmer  spot,  or  end,  on 
account  of  a  changed  condition  in  the  adhesiveness  of  the 
fluid,  and  so  a  spirit  level,  while  being  tested,  should  never  be 
touched  by  the  fingers,  nor  breathed  upon,  and  it  should  always 
be  protected  from  the  heat  of  the  sun,  or  of  artificial  lights. 
Care  should  also  be  taken  to  allow  sufficient  time  for  the  bubble 
to  settle  before  taking  a  reading. 


PART   II. 

ADVANCED    COURSE. 

ACOUSTICS,    HEAT,    ELECTRICITY    AND    MAGNETISM,    WITH 

AN   APPENDIX    ON    THE    DETERMINATION    OF 

GRAVITY,  AND  ON   THE  TORSION 

PENDULUM. 


ACOUSTICS. 


LIST   OF   EXPERIMENTS. 

PAGE 

I.  THE  SONOMETER 93 

Method  of  plucking  strings  ;  tuning  in  unison ;  exercises  on 
scale  formation. 

II.  THE  SCALE  OF  EQUAL  TEMPERAMENT 95 

Tuning  a  fork  to  any  desired  pitch;   counting  beats  with 
clock  or  stop  watch  ;  manner  of  bowing  forks. 

III.  LAWS  OF  THE  TRANSVERSE  VIBRATIONS  OF  STRINGS  .        .         98 

Tuning  a  string  to  a  fork. 

IV.  DETERMINATION  OF  PITCH 101 

(a)  Graphically,  as  in  the  case  of  a  tuning  fork. 

(b)  With  the  siren,  as  for  an  organ  pipe. 

(c)  By  a  calculation,  as  in  a  string. 

V.   LISSAJOUS'  OPTICAL  METHOD  OF  TUNING     ....        104 

(1)  By  mirrors. 

(2)  With  the  vibration  microscope. 

Mode  of  constructing  tuning  forks ;  and  how  they  may  be 
sharpened  or  flattened. 

VI.   HARMONIC  MOTION 108 

(1)  Composition  of  parallel  or  rectangular  vibrations. 

(2)  Blackburn's  pendulum. 

(3)  Wheatstone's  wave  apparatus. 

VII.  OVERTONES 114 

(1)  In  strings. 

(2)  In  forks. 

(3)  In  pipes. 


92  EXPERIMENTAL   PHYSICS. 


PAGE 


VIII.  THE  CHRONOGRAPH 117 

IX.  THE  CLOCK  FORK 118 

X.  MELDE'S  EXPERIMENTS  WITH  STRINGS 120 

XI.  HELMHOLTZ'S  APPARATUS  FOR  COMBINING  SIMPLE  TONES  .  122 

XII.   KONIG'S  ANALYZER 124 

The  principle  of  the  manometric  flame 

XIII.  THE  MANOMETRIC  FLAME 125 

General  application. 

XIV.  VELOCITY  OF  SOUND 129 

(1)  By  measurement  of  a  wave  length. 

(2)  By  resonance. 

(3)  Bosscha's  method. 

(4)  Kundt's  method. 

(5)  Regnault's  method. 

XV.   DOPPLER'S  PRINCIPLE    .        .        .        .        .        .        .        .        135 


EXPERIMENTS. 

I.     THE   SONOMETER. 

THIS  is  a  large  sounding  box,  a  little  more  than  a  meter  in 
length,  provided  with  a  bridge  at  each  end,  over  which  eight 
strings  are  stretched,  either  by  means  of  weights,  or  by  fixed 
pins  and  thumbscrews,  to  which  the  strings  are  attached.  The 
distance  between  the  bridges  is  usually  one  meter,  and  a  wooden 
scale  is  provided  alongside,  divided  into  millimeters.  Clamps 
with  weights  are  also  provided,  or  else  sliding  bridges,  to  enable 
one  to  set  any  desired  portion  of  a  string  in  vibration.  The 
method  of  using  the  instrument  is  to  obtain  steel  strings  about 
half  a  millimeter  in  diameter,  and,  stretching  them  over  the 
bridges,  to  tune  them  all  in  unison  with  any  chosen  pitch. 
This  may  be  done  very  accurately  (even  by  what  is  termed  an 
unmusical  ear)  after  a  little  practice,  by  listening  for  the  beats 
which  are  produced  when  two  sets  of  waves,  differing  slightly 
in  their  periodic  times,  are  sent  from  two  separate  sources 
simultaneously  to  the  ear.  The  tension  of  any  particular  string 
is  increased  or  diminished  until  it  begins  to  beat  with  the  one 
chosen  as  a  standard  ;  the  beats  will  be  at  first  very  rapid,  and 
cause  a  peculiar  rolling  effect  in  the  ear,  which  must  be  heard 
to  be  appreciated  ;  as  the  tension  is  properly  altered  they  be- 
come slower  and  slower,  and  finally  disappear.  The  two  strings 
are  then  in  unison,  and  the  same  process  is  repeated  with  the 
others,  until  all  give  the  same  note.  In  order  to  vibrate  a  string 
properly,  it  should  be  plucked  with  the  thumb  and  forefinger  at 
its  center  ;  or,  better  still,  the  ball  of  the  thumb  is  placed  on  the 

93 


94  EXPERIMENTAL   PHYSICS. 

middle  point  of  the  string,  the  direction  of  the  thumb  being 
nearly  at  right  angles  to  the  string,  and  then,  with  a  light  pres- 
sure, it  is  allowed  to  slip  off,  thus  producing  the  fundamental 
note,  comparatively  free  from  the  higher  harmonics.  Steel 
strings  are  preferable  to  others,  as  they  can  be  obtained  almost 
perfectly  uniform  throughout,  give  a  loud  tone,  and  last  indefi- 
nitely; but  they  must  be  plucked  properly,  or  else  they  give  most 
disagreeable  harmonics,  which  may  interfere  with  tuning. 
When  all  the  strings  are  in  unison,  the  major  diatonic  scale  may 
be  formed  by  choosing  the  first  open  string  as  the  fundamental 
or  starting  note,  and  then  adjusting  the  clamps  with  weights  on 
the  other  strings,  so  that  the  lengths  put  in  vibration  are  to  the 
length  of  the  fundamental  in  the  ratios  : 


This  simply  assumes  the  definition  of  the  major  diatonic  scale 
to  be  one  whose  notes  are  as  the  ratios  : 


and  also  that  the  pitch  of  a  string  varies  inversely  as  its  length. 
The  scale  of  equal  temperament  may  be  obtained  for  any  open 
string  by  taking  the  other  lengths  in  the  ratios  : 


The  ordinary  Minor  and  Harmonic  Minor  scales  may  be  had 
by  choosing  lengths  similar  to  the  major  diatonic ;  except  that 
for  the  former  the  ratio  -|  is  made  £  x  ff  =  f>  a  minor  third ;  and 
for  the  latter  |  is  made  f,  a  minor  third,  while  f  is  made 
f  xff  =  f,  a  minor  sixth. 

The  harmonic  scale  is  formed  by  taking  lengths  on  a  stretched 
string  inversely  proportional  to  the  natural  numbers  i,  2,  3,  4,  ... 
and  so  on. 

By  listening  carefully  a  great  many  of  the  notes  of  this  scale 
can  be  heard  in  the  case  of  any  freely  vibrating  string,  their 


THE   SCALE   OF   EQUAL  TEMPERAMENT.  95 

presence  there  being  theoretically  established  by  what  is  known 
as  Fourier's  theorem. 

The  notes  of  this  scale,  when  heard  as  auxiliaries  to  the 
fundamental  note  of  a  string,  are  usually  called  harmonics,  and 
are  of  the  greatest  importance  in  the  theory  of  Music. 

The  formation  of  the  scales  is  based  upon  the  assumption 
that  the  strings  are  all  of  the  same  material,  and  uniform 
throughout ;  and,  the  tension  of  all  being  the  same,  the  pitch 
of  any  portion  of  a  string  depends  inversely  on  its  length. 
See  Exp.  3. 


II.     THE   SCALE   OF   EQUAL   TEMPERAMENT. 

The  accurate  scale  of  physicists,  determined  by  the  series  of 
vibration,  ratios, 

i>   |,   t,   t>   I.   f>   ¥>   2, 

is  seldom  used  for  musical  purposes,  on  account  of  the  difficul- 
ties which  have  been  met  with  in  arranging  mechanism  for 
instruments  to  play  in  different  keys.  A  musical  instrument 
like  the  piano,  which  would  play  accurately,  according  to  the 
true  scale  of  the  physicist,  in  all  keys,  would  require  more  than 
twenty-five  notes  to  the  octave.  This  can  easily  be  seen  by 
writing  down  all  the  notes  necessary  for  playing  in  the  various 
keys,  and  then  striking  out  those  which  are  repeated.  Thus, 
if  we  commence  at  middle  C  on  the  piano,  we  can  obtain  the 
true  major  scale  by  tuning  eight  strings,  according  to  the  ratios 
expressing  pitch.  If  we  did  this  throughout  the  keyboard,  we 
should  have  a  complete  major  scale  (all  white  notes)  in  perfect 
order.  But  suppose  now  we  wish  to  play  perfect  intervals  in 
the  key  of  D :  it  is  evident  we  must  introduce  new  notes  which 
may  not  correspond  at  all  with  those  in  the  key  of  C;  and  as 
we  proceed  from  one  key  to  another  we  find  additional  notes 
necessary  for  each  change  of  key.  And  then,  if  we  take 
account  of  flats  and  sharps,  we  are  introducing  fresh  difficulties. 


96  EXPERIMENTAL   PHYSICS. 

Many  attempts  were  made  to  overcome  this  mechanical  diffi- 
culty by  reducing  the  number  of  notes  on  the  keyboard ;  but  the 
only  method  which  allowed  playing  in  all  the  keys  was  the  one 
finally  adopted.  In  the  scale  of  equal  temperament  the  octave 
was  defined  to  be  twelve  semitones ;  and,  as  the  pitch  of  the 
octave  is  twice  that  of  the  fundamental,  a  scale  of  thirteen  notes 
in  the  octave  was  chosen,  with  twelve  intervals,  the  pitch  of  each 
being  obtained  from  the  next  lower  by  multiplying  by  the  twelfth 
root  of  two,  which  corresponds  to  the  interval  called  a  semitone. 

In  most  stringed  instruments,  the  scales  are  approximately 
scales  of  equal  temperament,  although  in  the  case  of  the  piano 
more  latitude  is  allowed  the  tuner,  who  generally  makes  some 
intervals  a  little  greater  than  others,  being  guided  more  by  his 
individual  taste  than  by  the  factor  which  is  represented  mathe- 
matically by  ty~2. 

One  may  easily  construct  a  scale  of  equal  temperament  for 
any  fundamental  C  note  by  using  a  tonometer  or  series  of 
forks,  differing  from  one  another  by  8  or  16  vibrations  per 
second.  The  set  used  for  experimental  purposes  generally 
commences  with  256  or  512  v.s.,*  and  runs  up  to  1024.  For 
example,  to  construct  the  tempered  scale  commencing  with  512, 
one  would  calculate  the  vibration  numbers  corresponding  to 
the  other  twelve  notes  by  using  the  factor  ^5.  The  complete 
scale  would  then  be  given  by  the  numbers,  512,  542.4,  574.7, 
608.9,645.1,  683.4,  724.1,  767.1,  812.7,  861.1,  912.3,966.5,  1024. 

To  obtain  the  vibration  number  542.4,  two  forks  of  the  tonom- 
eter are  chosen,  536  and  544,  and  these,  if  put  in  vibration  to- 
gether, give  four  beats  per  second ;  if  now  the  pitch  of  544  be 

*  Note  on  Pitch  Notation,  —  Confusion  often  arises  in  the  calculation  of  pitch  from 
the  fact  that  the  English  use  the  term  -vibration  in  the  same  sense  as  the  complete 
vibration  of  a  pendulum,  to  and  fro,  while  the  French  System  takes  it  to  be  a  simple 
vibration  ;  so  that  a  French  fork  numbered  512  would  be  the  same  as  an  English 
256.  The  difficulty  is  overcome  by  marking  v.s.,  or  v.d.,  after  the  pitch,  as  is  done 
by  Konig  on  his  standard  forks,  to  indicate  vibrations  simples  (simple  vibrations), 
and  vibrations  doubles  (complete  vibrations).  We  have  adhered  to  this  in  the 
present  course. 


THE   SCALE   OF   EQUAL   TEMPERAMENT.  97 

altered  by  placing  small  pellets  of  wax  on  the  prongs  so  that  it 
moves  more  slowly,  it  is  evident  that  its  pitch  can  be  reduced  to 
the  required  number,  542.4.  In  other  words,  when  the  loaded 
fork  gives,  with  536,  3.2  beats  per  second,  or  32  beats  in  10 
seconds,  then  the  former  is  vibrating  542.4  times  per  second. 
The  process  then  becomes  one  of  counting  beats  with  a  clock 
or  stop  watch,  and  loading  the  forks  with  wax  until  the  necessary 
numbers  of  beats  are  obtained  in  each  case. 
The  following  precautions  should  be  taken  : 

1.  When  vibrating,  the  prongs  of  a  tuning  fork  move  out  and 
in,  together:  so,  if  one  sets  a  fork  in  vibration  with  a  bow,  it 
should  be  drawn  across  the  upper  end  of  one  prong  only,  and  in 
such  a  way  that  the  plane  of  the  horse  hair  is  nearly  in  the 
plane  containing  the  two  ends  of  the  prongs.     If  inclined  at  an 
angle,  a  pure  note  is  not  obtained,  and  the  horse  hair  is  very 
liable  to  get  torn.     The  fork  may  be  put  in  vibration  by  bowing 
at  the  side  of  the  prongs  ;  but  in  that  case  the  fundamental  tone 
of  the  fork  is  often  accompanied  by  a  disagreeable  high  note. 
SeeExp.  7(3). 

2.  In  counting  the  beats,  the  beginner  is  very  apt  to  count 
the  maximum  from  which  he  starts  as  one :    a  little  practice 
soon  enables  him  to  avoid  this. 

As  large  a  number  as  possible  should  be  counted  in  every 
case  to  minimize  the  error ;  usually  the  counting  can  be  carried 
on  for  thirty  or  forty  seconds,  except  in  the  case  of  high  forks 
which  have  small  amplitudes  and  soon  stop  vibrating. 

3.  In  loading  the  forks,  when  very  slight  differences  are  de- 
sired, it  will  be  found  more  convenient  to  move  the  pellets  of 
wax  up  or  down  the  prongs,  instead  of  increasing  or  diminishing 
the  loads  themselves. 

4.  Tuning  forks  are  always  made  of  steel  and  change  rapidly 
if  rusted.     Care  must  then  be  always  taken  of  them  by  the 
experimenter ;  and  when  an  experiment  is  finished  they  should 
be  carefully  cleaned  with  a  soft  rag  moistened   with  alcohol, 
dried,  and  put  away  in  a  dry  place. 


98  EXPERIMENTAL   PHYSICS. 

III.     LAWS  OF   THE   TRANSVERSE  VIBRATIONS  OF   STRINGS. 

If  a  string  of  any  material  be  stretched  beyond  its  natural 
length  between  two  fixed  points  and  then  be  disturbed  from 
its  position  of  equilibrium,  it  vibrates  to  and  fro,  and  emits 
certain  musical  notes,  of  which  one  is  much  more  prominent  than 
any  other :  this  is  the  lowest  or  fundamental  note  of  the  open 
string,  and  its  pitch  or  vibration  number  is  usually  spoken  of  as 
the  pitch  of  the  string.  This  pitch  will  evidently  depend  on  the 
length  of  the  string,  its  diameter,  tension,  and  the  material  of 
which  it  is  made ;  and  a  relation  connecting  these  elements  can 
be  found  theoretically  in  the  following  manner : 

Let  the  string  be  stretched  between  two  fixed  points  O  and 
A,  Fig.  i,  with  a  tension  T.  And  suppose  that  it  is  uniform 
throughout,  of  radius  r,  density  p,  and  length  /. 

Then,  if  disturbed  slightly,  and  the  motion  be  referred  to 
three  rectangular  axes  at  O,  of  which  OA  is  the  axis  of  x,  it 
c 


o^-- — 


Fig.  1. 

is  obvious  that  any  element  such  as  PQ  whose  mass  is  pds  will 
have  motions  parallel  to  OB  and  to  OC.  The  equations  of 
motion  of  this  element,  on  the  hypothesis  that  the  tension 
remains  constant  and  that  the  length  ds  is  equal  to  its  pro- 
jection dx,  will  be 


THE    TRANSVERSE   VIBRATIONS   OF    STRINGS.  99 

the  coordinates  of  the  point  P  being  x,  y,  z.     For,  since  the  ten- 
sion at  P   along  the   tangent  is    T,  the  tensions  along  lines 

through  P,  parallel  to  OB  and  OC,  will  be  T-j-y  T-j-  ;  and  at 
Q,  in  the  same  directions,  the  tensions  will  be 

^dy      d(rrdy\.         ,  ^dz      d  (  ~dz\  . 

T^  +  -r(  T^r)ds,  and  T-J-  +  -J-I  T-T)ds. 

ds      ds\    dsj  ds      ds\    dsj 

These  become 

d*y_T     d?y_ 
dt*~  p'  dx* 


since  ds  —  dx  ;  and  they  express  analytically  the  fact  that  the 
string  may  move  in  the  plane  of  xz,  or  in  the  plane  of  xy,  or  it 
may  have  both  motions  combined. 
The  solution  of  one  gives 

y  =f(x  -  at)  +  F(x  +  «')» 

where  a2  =  —• 
P 

This  represents  the  transmission  of  arbitrary  forms  along  the 
string  with  velocity  a,  where  ar  =  2  /,  T  being  the  periodic  time 
of  vibration  of  the  fundamental  note. 

Hence  T=-  =  2rl\l—, 

a  ^Tg 

where  d  is  the  specific  gravity  of  the  string  referred  to  water  as 

unity,  and  p  therefore  equal  to  ^—  ,  the  cross-section   of   the 

g 
string  being  circular.     Therefore  the  pitch  of  the  fundamental 

note  of  the  string,  which  is  equal  to  -,  becomes 


100  EXPERIMENTAL   PHYSICS. 

In  this  formula  are  included  all  the  laws  of  the  transverse 
vibrations  of  strings.  It  shews  us  that  the  pitch  varies  inversely 
as  the  length,  and  radius  of  the  string,  directly  as  the  square 
root  of  the  stretching  force,  and  inversely  as  the  square  root 
of  the  density.  All  these  facts  may  be  verified  experimentally 
by  varying  the  elements  of  the  string;  and  different  experi- 
ments can  easily  be  arranged  to  prove  any  particular  law.  The 
most  useful  and  instructive  exercise  for  the  student,  however, 
is  to  stretch  a  string  over  such  a  sonometer,  as  is  shewn  in  Fig.  2. 


Fig.  2. 

The  string  is  fixed  at  one  end,  passes  over  two  bridges  which 
determine  the  two  fixed  points  of  the  previous  theory,  and  then, 
running  over  a  pulley,  is  kept  stretched  by  means  of  known 
weights.  Its  mean  radius  is  determined  with  the  wire  gauge, 
its  length  is  measured,  or  observed  directly  on  the  scale  of  the 
sonometer,  and  its  specific  gravity  may  be  either  taken  from 
tables  or  found  in  the  usual  way.  Then  the  pitch  is  calculated 
by  the  formula  and  verified  by  means  of  a  standard  fork. 

This  latter  determination  is  sometimes  difficult  on  account  of 
the  peculiar  monotonous  tone  of  the  fork,  which  leads  one  to 
imagine  that  it  is  an  octave  lower  than  its  real  pitch,  as  well  as 
the  presence  of  loud  overtones  in  the  string  (especially  if  made 
of  steel)  which  an  unmusical  ear  often  chooses  instead  of  the 
fundamental. 

The  easiest  method  of  using  the  fork  is  to  move  the  sliding 
bridge,  usually  provided  with  the  sonometer,  along  until  it  cuts 
off  a  portion  of  the  string  which  will  vibrate  in  unison  with 
the  fork  ;  then  the  pitch  of  the  whole  string  will  be  to  that  of  the 


DETERMINATION  OF   PITCH.  IOI 

fork  in  the  inverse  ratio  of  the  length  of  the  open  string  to  the 
part  cut  off  by  the  bridge.  The  fork  should  be  chosen  of  a 
suitable  pitch  so  that  the  portion  of  the  string  cut  off  by  the 
bridge  may  not  be  too  small. 

If  the  sonometer  is  provided  with  two  sets  of  pins  and  pulleys, 
the  experiment  may  be  modified  by  stretching  two  strings  of 
the  same  material,  length,  and  diameter  with  different  weights  ; 
or  by  keeping  the  material,  length,  tension  constant,  and  vary- 
ing the  diameters. 

The  most  useful  strings  for  experimental  purposes  are  the 
fine  steel  E  strings  used  for  the  mandolin  and  guitar.  Copper 
or  brass  ones  may  also  be  used,  as  they  can  be  obtained  in  all 
sizes,  are  comparatively  uniform  throughout,  and  require  little 
tension  to  give  a  distinct  note ;  they  are,  however,  if  of  copper, 
apt  to  stretch  and  break.  Gut  strings  can  be  used,  but  they 
are  hygroscopic,  and  it  is  difficult  to  get  them  uniform  through- 
out their  length. 


IV.     DETERMINATION   OF   PITCH. 

(a)  Forks. 

The  method  employed  for  determining  the  pitch  of  a  tuning 
fork  is  graphical.  A  slight  style  or  writer  of  inappreciable 
weight  is  attached  to  the  fork  with  wax;  the  fork  being  so 
arranged  that  the  style  just  rests  against  a  cylinder  turning 
at  a  known  rate,  and  covered  with  a  smoked  paper.  As  the 
cylinder  is  turned  the  writer,  owing  to  the  vibrations  of  the 
fork,  makes  a  wavy  line  on  the  paper,  and  the  number  of 
these  vibrations  made  in  a  given  time  can  then  be  counted,  and 
the  pitch  of  the  fork  found. 

Generally,  a  standard  fork  is  placed  alongside  of  the  fork 
whose  pitch  is  to  be  found,  and  a  comparison  made  between  the 
two  tracings  on  smoked  paper ;  thus  rendering  the  determina- 
tion independent  of  the  rate  at  which  the  cylinder  is  turning  if 
the  points  of  the  two  styles  are  side  by  side.  The  style  may  be 


102 


EXPERIMENTAL   PHYSICS. 


made  of  a  bristle  or  stiff  hair,  or  of  thin  sheet  brass ;  but  in  the 
last  case  a  small  correction  will  have  to  be  made  unless  the  fork 
is  very  large. 

(b)   Organ  Pipes. 

The  pitch  of  an  organ  pipe  may  be  found  most  readily  and 
simply  by  comparison  with  a  fork,  which  may  be  chosen  so  as 
to  give  a  small  number  of  beats  per  second.  Care  must  be 
taken,  however,  that  the  fundamental  note  of  the  pipe  and  not 


Fig.  3. 

one  of  the  overtones  is  chosen.  Forks  with  sliding  weights 
to  alter  their  pitch  are  best  adapted  for  this  purpose. 

One  may  also  find  the  pitch  of  a  pipe  by  means  of  Helm- 
holtzs  siren,  which  is  shewn  in  Fig.  3. 

The  siren  is  attached  to  a  bellows  and  driven  at  constant 
speed.  Counters,  shewn  in  the  figure,  are  attached  to  the  ver- 
tical axis  of  the  instrument  and  enable  one  to  observe  the 
number  of  vibrations  in  any  given  time.  To  determine  the 


DETERMINATION   OF   PITCH.  103 

pitch  of  a  pipe,  the  discs  are  driven  until  the  note  emitted  by 
the  siren  is  in  unison  with  it,  and  if  the  note  is  maintained  for 
some  seconds  without  variation,  the  time  and  corresponding 
number  of  vibrations  can  be  found.  The  method  is  tedious, 
and,  unless  great  care  is  taken  to  maintain  a  constant  pres- 
sure in  the  air  chest  of  the  bellows,  no  accurate  result  can  be 
obtained.  If  the  axis  carrying  the  two  discs  is  driven  by  a 
small  electromotor,  thus  making  the  driving  power  independ- 
ent of  the  note,  better  results  may  be  obtained ;  the  instru- 
ment is,  however,  much  better  suited  for  illustrative  purposes 
on  the  lecture  table,  and  is  not  strictly  a  measuring  instrument. 
It  may  be  mentioned  here  that  the  simplest  form  of  siren,  and 
one  that  may  be  used  to  some  advantage  for  determining  pitch, 
is  a  circular  plate  of  brass  with  small  openings  symmetrically 
arranged.  When  driven  uniformly  by  any  rotation  apparatus, 
and  air,  blown  through  a  small  orifice,  is  directed  against  the 
openings  perpendicular  to  the  face  of  the  plate,  a  note  is  pro- 
duced whose  pitch  can  easily  be  found  from  the  number  of 
holes  and  the  time  of  rotation.  Such  a  plate  provided  with 
concentric  rings  of  holes  properly  arranged  may  be  used  for 
lecture  purposes  to  shew  the  formation  of  scales  and  combina- 
tions of  notes,  and  has  the  advantage  over  all  other  forms  of 
sirens  on  account  of  its  perfect  simplicity. 

(c)    Strings. 

The  pitch  of  a  stretched  string  is  found  in  the  manner  already 
explained,  by  comparison  with  a  standard  fork  directly,  or  by 
calculation  from  the  elements  of  the  string  where  they  are 
known. 

When  the  fork  chosen  is  not  near  the  string  in  pitch,  or 
where  it  is  not  possible  to  obtain  a  fork  close  enough,  then  the 
sliding  bridge  may  be  used  on  the  string,  and  a  portion  cut  off 
which  vibrates  in  unison  with  the  standard  :  the  pitch  of  the 
the  open  string  can  then  be  inferred. 

For  very  low  or  very  high  sounds,  some  difficulty  is  always 
experienced  in  determining  pitch. 


104  EXPERIMENTAL   PHYSICS. 

In  the  case  of  low  sounds  a  siren  may  be  used,  but  the  diffi- 
culty then  is  to  maintain  a  constant  pressure  in  the  air  chest. 
The  method  devised  by  Savart  may  be  used  both  for  very  low 
and  very  high  notes.  A  wheel  with  a  large  number  of  teeth  is 
driven  uniformly  at  any  desired  speed,  and  a  small  strip  of  stiff 
cardboard  or  metal  is  held  with  an  edge  just  touching  the  teeth ; 
the  noise,  which  is  heard  at  first  when  the  wheel  rotates  slowly, 
gradually  develops  into  a  decided  musical  note  that  becomes 
higher  as  the  speed  increases,  the  upper  limit  depending  only  on 
the  rate  at  which  it  is  possible  to  drive  the  wheel.  When  used 
to  determine  pitch,  its  note  is  made  identical  with  the  one  to  be 
observed,  and  the  pitch  calculated  from  the  speed  and  the 
number  of  teeth.  Conversely,  the  speed  of  a  rapidly  rotating 
wheel  or  disc  may  be  best  inferred  from  the  note  produced  by 
some  attachment  that  makes  periodic  sounds  which  change  to  a 
musical  note  under  high  speed.  This  method  was  used  by 
Wheatstone  in  his  measurement  of  the  velocity  of  electricity, 
and  is  of  some  importance. 

For  very  high  notes,  the  sonometer  with  a  sliding  bridge,  and 
a  very  tightly  strung  fine  steel  wire  may  also  be  used  to  advan- 
tage, and  will  give  fairly  accurate  measurements  of  pitch  as  high 
as  10,000  vibrations  per  second. 

Steel  cylinders  are  also  made,  to  furnish  standards  of  pitch 
for  very  high  notes,  and  to  test  the  upper  limit  of  audibility. 
They  range  from  8192  vibrations  to  40,000,  and  are  usually 
made  to  give  the  notes  of  the  major  diatonic  scale.  By  means  of 
them  one  may  locate  any  particular  high  note  within  a  few  hun- 
dred vibrations,  by  the  unassisted  ear. 

V.    LISSAJOUS'   OPTICAL  METHOD   OF   TUNING. 

(i)     With  mirrors  attached  to  the  forks. 

Instead  of  tuning  by  means  of  the  ear,  another  method, 
devised  by  Lissajous,  is  used  for  the  accurate  comparison  of 
forks  which  may  be  either  close  to  one  another  in  pitch,  or  very 


LISSAJOUS1  OPTICAL   METHOD   OF  TUNING.  105 

far  apart.  Mirrors  are  fixed  rigidly  to  the  prongs  of  the  forks, 
which  are  so  arranged  that  they  can  vibrate  in  two  planes  at 
right  angles  to  one  another ;  and  a  beam  of  light  from  a  small 
luminous  source  is  allowed  to  fall  in  succession  on  the  two 
mirrors,  and  then  received  on  a  screen,  viewed  directly  by  the 
eye,  or  allowed  to  pass  into  a  small  telescope  focussed  so  that 
the  image  of  the  luminous  source  is  clear  and  sharp.  On 
vibrating  the  forks,  the  image  in  the  telescope  is  seen  to  pass 
through  various  forms  owing  to  the  compound  motion  ;  and,  by 
loading  the  forks  until  certain  known  forms  are  obtained,  the 
ratio  of  the  two  vibration  numbers  can  be  inferred. 

Theory.  —  When  the  two  forks  vibrate,  their  motions  are 
periodic,  and  the  displacement  of  the  spot  of  light  seen  after 
reflection  from  the  two  mirrors  will  therefore  be  equivalent  to 
two  displacements  obtained  by  solving  the  equations  of  motion : 


x=a  COS27T— 


These  give 


Where  a  and  b  are  the  amplitudes  of  vibration,  Tand  T  the 
periodic  times,  and  d  a  constant,  called  the  phase,  which  depends 
on  the  initial  circumstances  of  motion.  On  eliminating  t,  we 
shall  get  an  equation  to  a  curve,  which  is  the  form  seen  in  the 
telescope. 

i.    If  T=  T',  the  forks  will  be  in  unison,  and  the  curve  seen 
will  have  the  form 


—  —r- 
tr      ao 


io6 


EXPERIMENTAL   PHYSICS. 


which  represents  an  ellipse  or  straight  line,  as  shewn  in  (a)  Fig.  4, 
according  to  the  values  of  S,  determined  by  the  initial  conditions 
of  vibration. 

If  T  differs  slightly  from  T',  then,  whatever  be  the  initial 
form,  the  curve  begins  to  change,  and  runs  through  all  the 
forms  shewn  ;  on  loading  the  forks  properly  with  small  pellets 
of  wax  the  change  takes  place  more  slowly,  until  when  T 
becomes  equal  to  T'  the  figure  initially  obtained  preserves  its 
form  throughout  the  motion. 

2.  If  T'  =  2  T,  then,  on  eliminating  /  as  before,  a  complicated 
equation  is  obtained,  which  represents  the  curves  shewn  in  (b). 


(Os) 


Fig.  4. 


In  a  particular  case,  where  S  =  J,  the  equation  reduces  to  a 
parabola. 

3.  The  figures  in  (c}  and  (d)  shew  the  forms  obtained  when 
r'  =  3  T,  and  2  r  =  3  T. 

There  are  many  ways  of  drawing  these  curves  geometrically  ; 
and  an  excellent  exercise  for  the  student  is  to  take  a  rectangle 
whose  sides  represent  the  amplitudes  and  map  out  a  series  of 
points  from  the  relations  for  the  displacements,  for  any  chosen 


LISSAJOUS'   OPTICAL   METHOD   OF  TUNING.  IO/ 

ratio,  assuming  some  particular  value  for  8.  The  curve  for  that 
particular  phase  is  then  easily  drawn. 

(2)    TJie  vibration  microscope. 

For  the  more  practical  purpose  of  comparing  forks  with 
standards,  an  instrument  known  as  the  vibration  microscope  is 
used.  It  is  represented  in  Fig.  5.  A  standard  fork  of  known 
pitch,  which  can  usually  be  altered  by  sliding  weights,  carries 

C: 


Fig.    5. 

on  one  prong  a  small  lens,  whose  axis  is  perpendicular  to  the 
plane  of  vibration  of  the  fork ;  and  a  fixed  microscope  is 
arranged  independently  behind,  so  that  its  axis  can  be  placed 
in  the  prolongation  of  the  axis  of  the  lens.  The  lens  on  the 
fork  forms  the  objective  of  the  microscope.  The  fork  to  be 
examined  is  then  placed  so  that  its  plane  of  vibration  is  per- 
pendicular to  that  of  the  standard  ;  and  any  small  mark  on  it 
is  viewed  through  the  microscope,  which  thus  shews  an  image 


108  EXPERIMENTAL   PHYSICS. 

of  the  mark  going  through  a  compound  motion,  when  the 
two  forks  are  vibrating,  similar  to  that  obtained  with  two 
mirrors  and  a  spot  of  light.  The  pitch  of  the  fork  is  inferred 
as  in  the  former  experiment.  Practically,  this  furnishes  an 
exact  way  of  reproducing  a  standard  ;  by  filing  slightly  the  ends 
of  the  prongs  the  pitch  can  be  raised  as  much  as  one  pleases, 
and  if  it  becomes  too  high  a  slight  filing  on  the  inside  of 
the  prongs  lowers  it  ;  and  by  operating  in  this  way  until 
the  stationary  ellipse  or  straight  line  is  obtained,  a  standard 
may  be  made  in  a  short  time.  The  accuracy  of  the  method 
is  such  that  one  may  get  two  forks  so  close  in  pitch  that 
they  will  not  differ  from  one  another  by  more  than  one  beat 
in  a  minute,  a  thing  which  no  system  of  tuning  by  ear  could 
ever  accomplish. 

VI.     HARMONIC   MOTION. 

(i)    Composition  of  parallel  and  rectangular  vibrations. 
A  motion  represented  by  the  equation 


is  said  to  be  periodic  or  harmonic.  The  prong  of  a  tuning  fork 
gives  such  a  motion,  as  may  be  seen  graphically  by  arranging  a 
small  writer  on  the  end  of  one  prong  of  the  fork  ;  when  put  in 
vibration  and  drawn  so  that  the  writer  moves  uniformly  along  a 
piece  of  smoked  glass,  the  tracing  is  an  accurate  curve  of  sines. 

Two  parallel  or  rectangular  harmonic  motions  may  be  com- 
pounded, by  aid  of  two  tuning  forks,  with  the  apparatus  shewn 
in  Fig.  6. 

One  fork  is  fixed  in  position,  and  holds  on  one  prong  a  plate 
of  smoked  glass  which  vibrates  with  it  ;  on  the  other  prong  is  a 
small  sliding  weight  which  is  used  as  a  counterpoise  to  the  glass 
plate.  The  other  fork,  provided  with  a  small  writer,  is  arranged 
independently  so  that  it  can  be  drawn  backwards  while  the 


HARMONIC   MOTION. 


109 


writer  rests  gently  on  the  smoked  glass.  To  compound  two 
parallel  motions,  the  forks  are  so  placed  that  their  prongs  and 
the  plate  of  glass  are  all  parallel ;  the  position  for  angular  com- 


position is  shewn  in  the  figure.     Both  forks  are  set  vibrating, 
and  then  the  one  carrying  the  writer  is  drawn  along  as  uni- 


..  ..  ^        ^        ^        ^ 

J     \J     \J      \j      \J      \J      \J     V 


v/ 


_ 

\r\r 


Fig.  7. 


formly  as  possible,  and  the  resulting  curve  traced  on  the  glass 
shews  the  composition. 

These  curves  may  also  be  drawn  geometrically,  and,  as  in 


110  EXPERIMENTAL   PHYSICS. 

the  case  of  Lissajous'  curves,  the  drawing  is  a  good  training  for 
the  proper  appreciation  of  wave  motion. 

Figure  7  shews  two  compositions  of  parallel  motions.  The 
lower  represents  two  curves  of  sines  of  which  one  wave  length 
is  twice  the  other,  and  their  composition  is  also  shewn.  The 
upper  one  shews  the  composition  of  two  curves  whose  wave 
lengths  are  as  I  to  3. 

Figure  8  shews  the  composition  of  two  curves  described  by 
forks  whose  vibration  numbers  are  in  the  ratio  of  4  to  5. 

\     /\     /\     /\     /\     /\     /\     /\      /^ 

\7     V7     \^7     \y     \^7     \y     \s     \y 
/\    /N    /\    /\    /\    /\    /\    /~\    /\    /~\ 
7     \s     \s     VT     \I7     \y     \37     \37     \7     \7     \s 

-  ^  ~ 


Fig.  8. 

Figures  9,  10,  11  shew  compositions  arising  from  forks  of  1:2, 
i  :  3,  i  :  4  very  nearly.  One  is  exactly  in  the  ratio  i  :  4.  The 
curious  chainlike  appearance  presented  by  them  is  caused  by 
the  slight  difference  of  phase. 

Figure  12  shews  the  composition  of  two  vibrations  nearly 
rectangular. 

Figure  13  represents  to  the  eye  the  effect  produced  on  the 
ear  by  two  sets  of  waves  which  have  very  nearly  the  same 
periodic  time.  It  is,  in  fact,  a  graphical  representation  of  a  beat. 

(2)    Blackburn  '.r  pendulum. 

The  composition  of  two  motions  at  right  angles  to  one  another 
may  also  be  exhibited  by  means  of  a  very  simple  contrivance 
known  as  Blackburn's  pendulum.  It  is  shewn  in  Fig.  14. 

Three  wires  are  joined  together  at  D,  the  two  upper  ones 
being  fastened  to  a  horizontal  beam,  while  the  lower  carries  a 
heavy  sphere  to  which  a  small  writer  is  attached.  If  the  sphere 
be  moved  perpendicularly  to  the  plane  of  the  paper,  it  oscillates 
about  the  point  E,  its  time  of  oscillation  varying  as  the  square 


HARMONIC  MOTION. 


1  II 


Fig    9. 


Fig.  10. 


Fig.  11. 


Fig.   12. 


Fig.   13. 


EXPERIMENTAL  PHYSICS. 


root  of  EC,  C  being  the  center  of  the  sphere.  If  disturbed  in 
the  plane  of  the  paper,  it  oscillates  about  D,  with  a  different 
periodic  time.  The  lengths,  EC, 
DC,  can  be  adjusted  so  that  the 
times  of  oscillation  are  in  any 
desired  ratio.  If  then  the  sphere 
be  moved  indifferently  in  any  direc- 
tion, and  a  piece  of  smoked  glass 
be  placed  underneath  so  that  the 
writer  is  always  in  contact  with  it, 
a  closed  curve  will  be  traced  out, 
arising  from  the  compositions  of  the 
two  rectangular  motions. 

Figure  16  is  a  photograph  of  an 
actual  tracing  made  with  a  rough 
pendulum  consisting  of  an  iron  ball 
The  writer  was  a  piece  of  thin  sheet 


Fig.  14. 


suspended  by  steel  wires, 
brass. 


Fig.   15. 


(3)    Wheatstones  wave  apparatus. 

This  shewn  in  Fig.  15  is  an  admirable  way  of  studying  the 


114  EXPERIMENTAL  PHYSICS. 

composition  of  parallel  and  rectangular  vibrations ;  also  for 
illustrating  the  general  phenomena  of  wave  motion. 

The  machine  has  a  number  of  stiff  iron  rods,  each  made  in 
the  form  of  a  cross  and  carrying  beads  at  the  ends  of  the  upper 
and  the  two  side  pieces.  These  rods  are  constrained  to  move 
in  one  plane.  Boards  are  supplied  with  the  apparatus,  which 
have  curves  of  sines  cut  along  their  edges,  and  are  made  in 
pairs  so  that  the  curves  fit  one  another.  The  iron  rods  are  so 
arranged  that  when  the  boards  are  placed  in  position  and 
pushed  through  the  machine,  the  motion  of  the  beads  shews 
the  propagation  of  the  wave  form,  which  may  be  either  simple 
or  complex.  If  the  horizontal  boards  are  used  alone,  the  beads 
have  rectilinear  motions,  which  indicate  the  propagation  of  a 
plane  horizontal  wave ;  if  the  vertical  boards  are  used,  plane 
vertical  waves,  resembling  somewhat  water  waves,  are  obtained  ; 
and,  by  combining  vertical  with  horizontal,  compound  waves 
like  those  of  light  may  be  formed. 

It  will  be  noticed  in  all  cases  that,  whatever  boards  or  com- 
binations be  used,  the  curve  on  top  is  the  resultant  of  the  two 
curves  shewn  at  either  side  by  the  sets  of  fixed  and  movable 
beads. 

VII.     OVERTONES. 

(i)'  In  strings  vibrating  freely,  the  complex  nature  of  the 
vibration,  as  determined  analytically  from  the  equations  of 
motion  by  the  aid  of  Fourier's  theorem,  may  be  seen  directly 
by  the  eye,  or  observed  by  the  ear  with  the  assistance  of  what 
are  called  resonators ;  these  are  hollow  globes  or  cylinders 
of  metal  or  glass,  constructed  of  various  sizes  so  that  the  air 
inside  vibrates  in  sympathy  with  certain  corresponding  notes, 
and  thereby  reinforcing  them  renders  it  possible  for  a  person 
to  hear  tones  which,  without  the  resonators,  would  not  be 
audible.  They  are  usually  made  in  a  series  corresponding  to 
the  notes  of  the  harmonic  scale,  and  may  be  either  fixed  or 
variable.  Figure  17  represents  two  kinds,  one  fixed  and  spherical 


OVERTONES.  1 1 5 

in  shape,  the  other  cylindrical  and  capable  of  adjustment  so 
as  to  be  used  for  several  tones. 

To  use  them  properly,  the  smaller  end  is  either  placed 
directly  in  the  aperture  of  the  ear  or  connected  with  it  by 
means  of  a  short  rubber  tube ;  the  other  end  being  then 
brought  into  the  vicinity  of  the  complex  note  to  be  examined. 
Success  in  the  experiment  is  best  attained  by  closing,  at  the 
same  time,  the  aperture  of  the  other  ear.  To  obtain  good 
results  from  a  string,  choose  a  steel  one  about  six  feet  long 
and  stretch  it  between  two  fixed  supports  until  it  gives  a  pitch 
of  256  v.s.  With  the  series  of  resonators  corresponding  to 
this  pitch  as  fundamental,  it  is  possible  to  recognize  eighteen 
or  twenty  pure  overtones  of  the  harmonic  scale.  In  fact,  after 


Fig.   17- 

a  little  practice  with  the  resonators,  they  may  be  discarded 
and  most  of  the  overtones  still  heard  by  the  unassisted  ear. 
With  steel  and  copper  strings,  the  third  harmonic  is  often 
louder  than  the  fundamental ;  and  it  is,  no  doubt,  owing  to  the 
presence  of  these  overtones  in  the  string  that  great  difficulty 
is  always  experienced  in  tuning  a  string  to  a  fork,  as  the  ear 
is  apt  to  choose,  instead  of  the  fundamental  note,  one  of  the 
more  prominent  overtones. 

(2)  In  the  case  of  organ  pipes,  the  complex  nature  of  the  note 
obtained  may  also  be  shewn  by  using  the  resonators  ;  but  the 
best  method  for  the  examination  of  the  notes  produced  by  both 
closed  and  open  pipes  is  that  of  the  manometric  flame,  in  con- 
nection with  the  resonator.  See  Exps.  12  and  13. 


Il6  EXPERIMENTAL   PHYSICS. 

(3)  The  timing  fork  is,  in  reality,  a  combination  of  two  steel 
rods  which  vibrate  in  unison  with  one  another ;  and  its  pitch 
varies  directly  as  the  thickness  of  the  prongs  (measured  in  the 
plane  of  vibration),  inversely  as  the  square  of  the  length,  and  is 
practically  independent  of  the  thickness  of  the  prongs  (measured 
perpendicularly  to  the  plane  of  vibration).  The  prongs  in  vibrat- 
ing, may  divide  into  segments,  just  like  a  string;  and  there- 
fore may  be  made  to  give  tones  superior  to  what  is  commonly 
called  the  pitch  of  the  fork.  These  tones,  however,  are  not  true 
harmonics :  the  first  one  being  nearly  six  times,  and  the  next 


Fig.   18. 

higher,  seventeen  times,  the  pitch  of  the  fork.  To  attain 
the  first,  the  fork  is  bowed  at  the  side,  about  half  way  down  ;  the 
second  may  be  obtained  by  bowing  at  a  point  one  third  of  the 
way  from  the  bottom.  Or  they  may  be  obtained  along  with 
the  fundamental  by  bowing  the  fork  first  in  the  usual  way  on 
top  and  then  along  the  sides.  Figure  18  shews  tracings  of  a 
fork  accompanied  by  its  superior  tones. 

The  lower  right-hand  tracing  is  made  by  the  fundamental 
with  its  first  superior  tone  ;  the  lower  left  hand  one  is  the  funda- 
mental with  the  second  superior  tone ;  and  the  upper  one  is  a 
combination  of  the  fundamental  with  the  first  two  superior 
tones. 


THE   CHRONOGRAPH. 


117 


VIII.     THE   CHRONOGRAPH. 

This  is  an  instrument  which  is  used  for  the  measurement  of 
small  intervals  of  time  by  means  of  an  electric  interrupter  and 
tuning  fork.  The  arrangement  is  shewn  in  Fig.  19. 

A  tuning  fork,  held  upright  in  a  rigid  support,  can  be  put  in 
vibration  by  a  current  of  electricity,  which  passes  along  one 
prong  of  the  fork,  through  a  small  contact  piece  that  acts  as  an 


Fig.  19. 


interrupter,  and  then  through  two  electro  magnets  situated  one 
on  each  side  of  the  fork,  as  shewn  in  the  figure.  To  the  end  of 
one  of  the  prongs  of  this  fork  is  attached  a  writer  of  brass  which 
rests  against  a  piece  of  smoked  paper,  so  arranged  in  a  roll  that 
it  can  be  drawn  along  continuously  by  the  handle  shewn  at  the 
upper  part  of  the  apparatus.  Two  small  writers,  one  on  each 
side  of  the  tuning-fork  writer,  act  as  interrupters,  when  the 
currents  passing  through  the  electro-magnets  which  carry  the 
writers  are  in  any  way  broken.  When  the  fork  is  running  elec- 


Il8  EXPERIMENTAL   PHYSICS. 

trically,  it  makes  a  wavy  line  on  the  smoked  paper  as  it  is  un- 
rolled, each  distance  from  crest  to  crest  corresponding  to  - 

n 
seconds,  where  n  is  the  pitch  of  the  fork  in  double  vibrations. 

The  two  interrupters  at  the  side  make  continuous  straight 
lines.  If  now  the  interrupters  are  arranged  in  a  circuit  which 
can  be  broken  so  as  to  indicate  the  beginning  and  the  end  of  any 
particular  phenomenon,  then,  if  the  interruptions  are  properly 
made  while  the  paper  is  moving  and  the  fork  vibrating,  marks 
will  be  visible  on  the  smoked  paper  where  the  tracers  of  the 
interrupters  have  departed  from  a  straight  line.  By  counting 
the  number  of  waves  between  these  marks  the  time  of  the  event 
is  found.  Instead  of  using  two  interrupters,  one  only  may  be 
used ;  although  in  using  both,  one  serves  as  a  check  upon  the 
other.  The  two  interrupters  can  also  be  arranged  in  separate 
circuits  independent  of  one  another,  while  the  tuning-fork  cir- 
cuit is,  of  course,  independent  of  both. 

With  such  an  instrument  as  this,  and  with  due  care,  one  may 
measure  with  great  accuracy  the  time  of  vibration  of  a  body,  or 
the  time  of  a  falling  body,  or  any  such  interval  where  it  is  possi- 
ble to  form  electrical  circuits  with  the  interrupters. 

The  auxiliary  figure  represents  an  apparatus  for  smoking  the 
roll  of  paper  in  a  proper  manner.  A  hollow  brass  cylinder  is 
fixed  in  position  between  two  wooden  supports,  and  the  paper 
is  unrolled  from  one  grooved  wheel  to  another,  passing  around 
the  cylinder  as  shewn  in  the  figure  and  being  blackened  by  a  coal 
oil  lamp  which  is  placed  underneath  the  cylinder.  To  avoid 
burning  the  paper  the  cylinder  is  partially  filled  with  water. 

IX.     THE   CLOCK   FORK. 

This  (Fig.  20)  consists  of  a  fork  of  100  v.d.,  held  upright  in  a 
firm  support  and  connected  to  a  clockwork  movement  in  such 
a  manner  that  it  regulates  it  through  the  escapement  just 
as  a  pendulum  would  ;  but  it  receives  at  each  oscillation  a  small 
impulse  which  enables  it  to  keep  up  a  steady  vibration. 


THE   CLOCK   FORK.  1 19 

The  two  branches  of  the  fork  carry  sliding  weights  attached 
to  micrometer  screws  so  that  the  period  of  vibration  can  be 
regulated  with  great  precision.  In  addition,  one  of  the  prongs 
carries  the  objective  of  a  microscope  whose  ocular  is  placed  on 
an  independent  support ;  thus,  it  may  be  used  in  the  same  man- 


Fig.  20. 

ner  as  the  optical  comparator  of  Lissajous.      A  thermometer 
placed  between  the  prongs  of  the  fork  gives  the  temperature. 

The  instrument  gives  perfect  results  when  properly  regulated, 
and  enables  one  to  observe  the  effects  of  temperature  on  the 
period  of  vibration  of  a  tuning  fork  by  using  an  auxiliary  fork 
whose  temperature  is  gradually  increased,  and  noticing  the 


120  EXPERIMENTAL   PHYSICS. 

figures  obtained  with  the  aid  of  the  vibration  microscope.  It 
serves  also  to  study  in  a  similar  way  the  effect  of  reinforcement 
on  the  pitch  of  a  fork  by  observing  the  vibration  of  a  fork 
placed  on  supports  of  different  materials  and  forms.  It  may  be 
stated  here  that  the  ordinary  steel  forks  between  128  v.s.  and 
1024  v.s.  have  their  pitch  diminished  by  about  -§^§  for  each 
increase  of  temperature  of  i°  C.  This  coefficient  increases 
slightly  as  the  pitch  of  the  fork  increases,  and  also  changes 
considerably  with  the  material  of  the  fork. 


X.     MELDE'S   EXPERIMENTS. 

A  tuning  fork  has  a  string  attached  to  one  prong  and  stretched 
over  a  pulley  by  a  small  weight,  or  else  depending  vertically 
from  the  fork.  When  put  in  vibration,  the  fork  sends  waves 
along  the  string  which  will  be  transversal  or  longitudinal  accord- 
ing to  the  position  of  the  string  relatively  to  the  plane  of  vibra- 
tion of  the  fork.  The  string  then  divides  up  into  loops  and 
nodal  points  according  to  the  well-known  laws  of  stretched 
strings  explained  in  Exp.  3.  A  simple  experiment  for  trans- 
verse vibrations  is  to  alter  the  length  and  tension,  and  prove 
that  the  number  of  loops  obtained  varies  inversely  as  the  square 
root  of  the  tension.  The  fork  may  be  kept  vibrating  electri- 
cally by  using  an  ordinary  dry  or  mercurial  interrupter.  Strings 
of  various  kinds  may  be  used  to  prove  the  law  of  density :  silk 
cord,  fine  steel,  copper,  or  platinum  wires  give  good  results. 

The  complex  vibration  of  a  string  attached  to  two  forks  may 
be  shewn  by  means  of  the  apparatus  indicated  in  Fig.  21,  where 
the  forks  are  vibrated  electrically  and  the  string  is  stretched 
between  them. 

Curious  and  instructive  complex  motions  are  obtained  by 
using  a  silk  cord  about  a  millimeter  in  diameter  and  a  meter  in 
length  stretched  from  a  fork  (vibrating  electrically)  of  128  v.s. 
On  placing  the  string  so  that  it  makes  an  angle  of  45°  with  the 
fork  in  its  plane  of  vibration,  and  stretching  it  with  the  proper 


MELDE'S    EXPERIMENTS.  121 

weight  (which  can  easily  be  found  by  trial  or  by  calculation), 
one  sees  it  divide  up  into  two  portions  which  resemble  solid 
figures  whose  cross-sections  are  usually  parabolas  or  lemnis- 
cates,  and  in  the  center  is  a  plane. 

This  seems  to  arise  from  the  two  sets  of  waves  which  are 
travelling  along  the  string  with  velocities  in  the  ratio  i  :  2. 

Tyndalfs  experiment  with  the  luminous  wire  may  also  be 
shewn  by  using  an  electrically  vibrating  fork  and  a  platinum 
wire  a  millimeter  in  diameter  and  a  meter  long,  which  runs  over 
a  brass  pulley  and  is  stretched  by  the  proper  weight.  An 
independent  current  is  then  passed  along  the  wire  by  one  prong 


Fig.  21. 

and  the  brass  pulley  which  gives  good  contact ;  the  wire  being 
then  made  red  hot  by  a  current  of  10  or  15  amperes,  the  fork  is 
set  vibrating;  when  immediately  one  sees  the  loops  become 
dark,  owing  to  cooling  by  their  rapid  motion,  and  the  nodes 
grow  brighter.  More  current  is  then  turned  on  until  the  loops 
again  become  bright,  and  the  whole  outline  is  seen ;  the  wire  is 
then  very  apt  to  break  at  the  pulley,  or  at  a  nodal  point,  owing 
to  the  unequal  distribution  of  heat.  A  heavy  steel  wire  may 
also  be  used  for  this  purpose,  but  in  that  case  the  weights  must 
be  adjusted  after  it  becomes  red  hot,  on  account  of  its  losing  its 
elasticity.  No  doubt  the  best  results  would  be  obtained  by 
using  a  carbon  filament,  such  as  is  seen  in  an  incandescent 
lamp,  since  it  possesses  the  advantage  of  having  its  resistance 
diminished  as  the  temperature  is  increased. 


122 


EXPERIMENTAL   PHYSICS. 


XI.     HELMHOLTZ'S    APPARATUS    FOR    SYNTHESIS. 

This  is  shewn  in  the  adjoining  figure.  A  series  of  forks 
running  from  ut^  to  mts  are  arranged  in  the  same  electrical  cir- 
cuit with  an  interrupting  fork.  Resonators  are  placed  behind 
each  fork,  and  are  capable  of  small  adjustments  so  that  they 
may  be  moved  backwards  and  forwards. 


Fig.  22. 

Stoppers  are  provided  for  the  resonators  and  connected  by 
means  of  strings  with  a  keyboard  in  front,  in  such  a  way  that 
on  depressing  any  key  the  corresponding  fork  is  heard  vibrating 
loudly.  When  the  keys  are  all  down  and  the  stoppers  closed, 
all  that  is  heard  is  a  gentle  humming  sound. 


HELMHOLTZ'S   APPARATUS. 


I23 


The  object  of  the  apparatus  is  to  combine  the  fundamental 
note  ntz  with  any  of  the  harmonics  in  the  series,  and  observe 
the  quality  or  timbre  of  the  corresponding  tone.  It  does  not 
admit  of  accurate  comparison,  but  yet  affords  an  instructive 
exercise  for  the  student.  The  noise  of  the  interrupting  fork 
may  be  overcome  to  a  large  extent  by  dividing  the  current,  or 


Fig.  23. 


introducing  a  great  resistance  into  the  circuit ;  or  the  interrupt- 
ing apparatus,  which  is  quite  independent  of  the  other  part  of  the 
instrument,  may  be  carried  to  another  room,  and  the  circuit 
through  the  forks  completed  by  wires  leading  from  one  room  to 
the  other. 


124  EXPERIMENTAL  PHYSICS. 

XII.     KONIG'S   ANALYZER. 

This  is  the  reverse  of  the  apparatus  described  in  the  preced- 
ing experiment :  it  is  used  to  analyze  the  constituents  of  a 
compound  note.  The  general  appearance  of  the  instrument  is 
shewn  in  Fig.  23. 

The  principle  used  is  that  of  the  manometric  flame,  devised 
by  Konig,  combined  with  the  rotating  mirror  of  Wheatstone.  In 
Fig.  24  a  section  of  the  manometric  capsule  is  given.  A  small 
wooden  box  encloses  a  membrane  M,  which  is  stretched  between 
two  points  B  and  C  and  separates  an  air  chamber  from  a  gas 
chamber.  The  gas  is  supplied  at  G,  and  burns  at  F  with  a 
small  narrow  flame  about  half  an  inch  in  height.  The  air  from 
any  source  enters  at  A. 

In  the  analyzing  apparatus  a  series  of  these  manometric 
capsules  have  their  air  chambers  connected  with  the  resonators 

by    tubes    at    the    back    (not 
shewn    in    the    figure).      The 
f  tubes  seen  in  front  lead  from 

*  1 1    the  gas  chambers  to  the  large 

tube  at  the  bottom  which  sup- 
plies gas  to  them  all.  Stop- 
cocks are  provided  for  each  gas 
jet,  so  that  any  one  may  be 

Fig.  24. 

shut  off  if  necessary. 

In  order  to  examine  any  compound  note,  such  as  that  pro- 
duced by  an  open  pipe,  its  harmonics  are  calculated  from  the 
pitch  of  its  fundamental,  the  resonators  are  adjusted  for  these 
harmonics,  and  when  the  pipe  is  blown  and  held  near  the  front 
of  the  apparatus,  while  the  mirror  (shewn  at  the  right)  is 
turning,  certain  of  the  flames  will  be  seen  to  give  wavy  appear- 
ances in  the  mirror,  shewing  that  they  are  vibrating,  by  the 
intermediary  of  the  elastic  membranes,  in  sympathy  with  the 
air  in  certain  of  the  resonators. 

The  instrument  furnishes  a  useful  means  of  observing  over- 


K 


MANOMETRIC   FLAMES. 


125 


tones  from  any  source  of  sound  ;  but  it  must  be  adjusted  with 
considerable  care  in  order  to  perform  its  work  with  accuracy. 


XIII.     MANOMETRIC   FLAMES. 

The  principle  of  the  manometric  flame  may  also  be  used  for 
many  other  purposes  :  notably  for  observing  the  complexity  of 
the  vowel  sounds  or  of  notes  sung  by  the  human  voice,  for 


Fig.  25. 

shewing  interference  of  waves  of  sound,  and  for  the  general 
composition  of  wave  motions. 

Figure  25  shews  how  it  may  be  used  for  the  examination  of 
the  vowels  sung  according  to  any  chosen  pitch. 

The  manometric  capsule,  a  section  of  which  is  at  A,  is 
attached  by  means  of  a  tube  to  a  funnel-shaped  mouthpiece  for 
collecting  the  waves  of  sound.  The  rotating  mirror  is  shewn 
at  M. 

Figure  26  explains  the  manner  of  comparing  the  vibrations  of 
two  columns  of  air  in  organ  pipes  ;  and,  by  connecting  the  two 


126 


EXPERIMENTAL   PHYSICS. 


gas  tubes  leading  from  the  capsules  in  the  organ  pipes  with  the 
two  ends  of  a  T-shaped  tube,  a  single  gas  jet  can  be  obtained 
which  is  actuated  by  the  two  sets  of  vibrations  simultaneously. 
A  number  of  these  combinational  vibrations  are  represented 


Fig.  26. 

below:  they  have  been  obtained  in  the  manner  indicated,  and 
correspond  to  the  intervals  of  the  major  diatonic  scale. 

The  manometric  flames  may  also  be  employed  in  conjunction 
with  the  resonators  of  tuning  forks.  In  most  of  the  resonators 
supplied  by  Konig  with  his  forks,  there  is  a  small  brass  tube  at 
the  back  provided  with  a  stopper,  the  use  of  which  is  to  con- 


MANOMETRIC   FLAMES. 


127 


nect  the  resonator  with  a  manometric  flame  by  means  of  a  long 
rubber  tube  and  thus  observe  the  vibrations  of  the  air  within. 
Any  number  of  forks  may  thus  be  used  and  the  composition 


Fig.  27. 


of  their  vibrations    studied  by  connecting  their  resonators  all 
to  a  common  gas  jet. 

If  forks  forming  a  harmonic  scale  be  chosen,  one  may  com- 


128  EXPERIMENTAL   PHYSICS. 

bine  them  optically  and  see  the  effect  which  is  obtained  by  the 
ear  with  the  apparatus  of  Helmholtz. 

Perhaps  the  most  striking  experiment  is  that  of  Konig  for 


Fig.  28. 


shewing  interference.    The  apparatus  shewn  above  is  constructed 
in  accordance  with  the  principle  introduced  by  Herschel 


VELOCITY    OF    SOUND    IN    AIR. 


I29 


A  tuning  fork  sends  vibrations,  reinforced  by  a  resonator,  to  a 
tube  which,  between  its  extremities,  is  divided  into  two  branches, 
the  length  of  one  of  which  can,  by  being  drawn  out,  be  altered 
at  will.  The  vibrations  are  sent  to  two  independent  gas  jets, 
after  having  travelled  different  dis- 
tances, and  a  third  gas  jet  enables 
one  to  see  the  effect  of  compound- 
ing the  two  sets  of  waves.  The 
manometric  capsules  are  inserted 
between  the  gas  jets  and  the  tubes. 

The  instrument  admits  of  perfect 
adjustment,  and  when  complete 
interference  of  the  two  wave  mo-  Fig.  29. 

tions    takes    place    the    appearance 

of  the  three  flames,  as  seen  in  a  rotating  mirror,  is  as  shewn 
in  Fig.  29. 


XIV.     VELOCITY   OF   SOUND    IN    AIR. 

The  velocity  with  which  sound  is  propagated  through  any 
medium  will  evidently  depend  on  the  elasticity  and  density  of 
this  medium,  and  theoretically  it  may  be  calculated  from  the 
formula 


where  E  is  the  coefficient  of  elasticity  and  D  the  density. 

In  solids  E   and  D   can   be  found  experimentally,  and   the 
determination  of  the  velocity  presents  no  difficulty. 

For  liquids  the  above  formula  becomes 


where  p  is  the  density  of  mercury,  H  the  height  of  the  normal 
barometer,  K  the  coefficient  of  compressibility,  and  D  the  den- 
sity of  the  liquid  at  the  observed  temperature.  But  in  the  case 


130  EXPERIMENTAL   PHYSICS. 

of  air  or  any  gas  an  experimental  method  of  finding  V  directly 
is  preferable  on  account  of  the  difficulty  of  determining  accu- 
rately the  heat  constants  which  are  involved  in  E. 

A  method  that  at  first  sight  appears  simple  is  the  direct  one 
of  firing  a  gun  at  a  known  distance  from  an  observer,  who  esti- 
mates in  some  way  the  time  which  elapses  between  the  percep- 
tion of  light  and  that  of  the  sound ;  but  the  personal  errors  in 
the  appreciation  of  this  interval  seriously  interfere  with  the 
accuracy  of  the  result,  and  the  method  at  best  is  but  approxi- 
mate. 

Other  methods  are  given  here  to  illustrate  the  various  ways 
in  which  the  velocity  of  sound  in  air  may  be  measured  ;  modi- 
fications in  some  of  them  would,  no  doubt,  insure  greater  accu- 
racy. 

(i)    By  measurement  of  a  wave  length. 

We  know  that  when  a  periodic  disturbance,  for  example  that 
from  a  vibrating  fork,  is  propagated  through  the  air,  any  parti- 
cle of  this  air  is  in  the  same  state  of  vibration  at  successive 
times  separated  by  an  interval  T  which  is  therefore  called  the 
periodic  time  and  is  equal  to  — ,  where  N  is  the  pitch  in  double 

vibrations ;  and  the  wave  length  is  defined  to  be  the  distance 
which  the  wave  form  travels  in  this  time  T. 
Hence  we  have 

X  =  VT, 

if  V  be  the  velocity  of  propagation  and  X  the  wave  length.  If 
then  we  take  a  tuning  fork  and  measure  X  in  any  way,  we  can 
calculate  V. 

With  forks  of  very  low  pitch  and  consequently  of  great  wave 
length  this  can  be  done  with  some  degree  of  exactness  by  aid 
of  the  ear,  either  unassisted  or  with  a  resonator.  The  fork  is 
put  in  vibration  in  front  of  a  reflecting  surface,  and  the  observer 
chooses  points  in  the  surrounding  medium  where  maximum  or 
minimum  effects  are  produced  on  the  ear.  A  measurement  of 
the  distance  between  successive  maxima  or  minima  gives  half 


VELOCITY  OF   SOUND    IN    AIR.  131 

the  wave  length  ;  and  the  pitch  being  generally  given  accurately 
on  the  fork,  we  have 

V=  \N. 

An  improvement  on  this  method  might  be  made  by  using  reso- 
nators in  connection  with  manometric  flames. 

The  most  accurate  way  perhaps  to  estimate  the  wave  length 
of  a  fork  is  to  use  the  apparatus  shewn  in  Fig.  28,  and,  after 
obtaining  complete  interference,  to  measure  the  wave  length  by 
means  of  the  scale  attached  to  the  apparatus. 

Assuming  the  pitch  of  the  fork,  the  velocity  of  sound  can 
then  be  inferred.  The  same  apparatus  may  also  be  used  for 
comparing  the  velocities  of  sound  in  different  gases. 

(2)    By  resonance. 

If  a  hollow  cylinder,  open  at  one  end,  be  fitted  at  the  other 
with  a  tight  plug  and  piston  rod  which  can  be  moved  backwards 
or  forwards,  and  a  tuning  fork  be  put  in  vibration  and  held  with 
the  prongs  over  the  open  end,  it  will  be  found  that,  by  adjusting 
the  movable  plug,  a  point  is  reached  where  the  air  column 
within  the  cylinder  vibrates  in  sympathy  with  the  fork,  and  acts 
as  a  resonator.  In  this  case,  since  X  =  VT, 


where  /  is  the  length  of  the  vibrating  air  column  and  N  the 
pitch  of  the  fork  in  double  vibrations. 

The  method  is  not  exact,  a  considerable  error  arising  from  the 
size  of  the  open  end  of  the  cylinder.  Rayleigk's  correction  for 
this  gives 

F=4(/+r)  N 

where  r  is  the  radius  of  the  tube  supposed  cylindrical.  If  the 
tube  be  of  sufficient  length,  it  is  evident  that  a  series  of  maxi- 
mum reinforcements  may  be  found,  and  then 


2«  +   I 

where  n  is  zero  or  any  integer. 


132 


EXPERIMENTAL   PHYSICS. 


The  adjoining  figure  shews  how  the  experiment  may  be  tried. 
A  tube  AB  a  few  feet  in  length  is  attached  by  means  of  a  clamp 
at  B  to  another  tube  with  a  reservoir  R,  and  is  provided  with 
two  stopcocks  /,  /,  for  the  regulation  of  the  supply  of  water 
from  the  reservoir.  Adjustment  is  then  made  until  the  distance 
AL  gives  perfect  resonance. 
(3)  Basse/la's  method. 

This  method,   although  not  more  exact  than  the  preceding, 
involves  a  new  idea,  that  of  noting  by  ear   the  coincidence  of 

two  sounds.  The  apparatus,  de- 
vised by  Konig,  consists  of  two 
boxes  provided  with  sounders 
and  electro-magnets,  which  are 
inserted  in  an  electrical  circuit 
with  a  mercurial  interrupter, 
whose  period  can  be  regulated 
by  means  of  adjustable  masses. 
The  axis  of  the  interrupter  car- 
ries a  mirror,  and  a  fixed  tuning 
fork  of  40  v.cl.  also  carrying  a 
mirror  is  so  situated  that  the 
image  of  a  silvered  bead  is  seen 
by  reflection  from  the  two  mir- 
rors. The  fork  and  interrupter 
vibrate  in  perpendicular  planes  ; 
and  the  time  of  interruption  is 
found  from  the  figure  given  by  the  image  of  the  bead.  It  is 
usually  made  one  tenth,  or  one  eighth  of  a  second,  the  corre- 
sponding figures  having  then  four  or  five  loops.  When  the 
adjustment  to  get  the  correct  time  of  interruption  has  been 
made  the  sounders  are  set  in  motion,  and  of  course  give 
simultaneous  raps  when  heard  by  an  ear  near  them ;  one  of 
them  is  then  carried  away  by  the  observer,  who  then  hears 
the  raps  coming  alternately  and  then  together  again.  He 
notes  the  distance  at  which  they  appear  coincident :  and  this 


v 

Fig.   30. 


VELOCITY   OF   SOUND   IN  AIR.  133 

distance,  divided  by  the  period  of  interruption,  must  give  the 
velocity  of  sound. 

If  then  V=  velocity  of  sound  in  air  at  zero, 
t  —  mean  temperature  of  the  air, 
T=  period  of  interruption, 
d  =  distance  of  coincidence, 
d 


then 


where  «  =  . 003665,  and  /  is  the  mean  temperature  of  the  air. 
By  taking  two  or  three  coincidences  and  varying  the  period  of 
interruption  fairly  good  results  may  be  obtained  ;  but  the  error 
in  estimating  a  single  coincidence  by  the  ear,  however  skilled  it 
may  be,  may  amount  to  as.  much  as  one  foot :  so  that  the  velocity 
is  liable  to  be  in  error  eight  or  ten  feet  per  second. 

(4)    Kundi 's  method. 

This  enables  one  to  compare  velocities  in  gases  with  one 
another ;  and  also  the  velocity  in  air  or  any  gas  with  that  in  a 
solid.  It  is  susceptible  of  great  accuracy  and  the  apparatus  is 
simple  and  easily  arranged. 

A  glass  tube  AB  rests  freely  on  two  supports,  and  at  one  end 
is  a  movable  plunger  C,  and  at  the  other  end  a  cork  Bt  through 


Fig.  31. 

which  passes  a  metal  rod  DF,  fixed  at  its  center  E,  and  having 
at  D  a  small  circular  disc  which  fits  the  tube  loosely.  If  DFbc 
put  in  longitudinal  vibration  and  the  length  CD  properly  ad- 
justed by  the  plunger,  it  will  be  found  that  nodes  and  loops 
are  formed  in  the  air  inside ;  and  if  lycopodium  be  strewn  lightly 


134 


EXPERIMENTAL   PHYSICS. 


within  the  tube  it  will  collect  and  form  little  heaps  where  the 
vibrating  air  is  at  rest ;  and  it  is  evident  that  the  rate  of  propa- 
gation of  sound  in  air  is  to  that  in  the  metal  DFas  the  distance 
between  two  loops  (or  which  is  the  same  thing,  two  nodes)  is  to 

the  length  of  the  rod.  If 
then  v  is  the  velocity  of 
sound  in  air,  V  that  in  the 
metal,  /  the  distance  be- 
tween the  little  loops  of 
powder,  L  the  length  of  the 
rod, 


By  placing  a  similar  tube 
in  connection  with  the  other 
end  F,  and  using  another 
gas  in  it,  the  velocities  of 
sound  in  air  and  in  the  other 
gas  will  evidently  be  to  one 
another  as  the  distances  be- 
tween the  respective  nodal 
points. 

(5)  Regnaulfs  method. 
By  far  the  most  accurate 
method  is  that  devised  by 
Regnault,  who  determined 
by  direct  measurement  of 
distance  and  time  the  ve- 
locity of  propagation  of 

sound  in  tubes  of  various  diameters ;  and  the  measurement  of 
time  was  made  independent  of  the  observer,  thus  eliminating 
the  personal  error.  A  modified  form  of  the  apparatus  used  by 
him,  which  enables  one  to  perform  the  experiment  in  a  limited 
space,  is  shewn  in  Fig.  32. 


Fig.  32. 


DOPPLER'S   PRINCIPLE.  135 

A  series  of  tubes  is  arranged,  forming  one  long  tube,  and  an 
electro-magnetic  arrangement  at  one  end  is  attached  to  a  pistol 
which,  when  discharged,  produces  a  disturbance  that  travels  to 
the  other  end  and  there  causes  a  mark  to  be  made  on  some  re- 
cording apparatus,  or  can,  if  necessary,  be  reflected,  return  and 
again  travel  to  the  other  end,  and  so  on.  The  pistol  is  also 
connected  with  the  recording  apparatus,  which  may  be  either  a 
cylinder  with  tuning-fork  attachment  or  the  chronograph  of 
Exp.  8. 

The  distance  along  the  axis  of  the  tube  can  easily  be 
measured  :  in  the  apparatus  constructed  by  Konig  it  is  about 
30  meters. 

Besides  being  used  for  measuring  the  velocity  of  sound,  this 
instrument  enables  one  to  investigate  all  the  phenomena  of 
reflected  sound  waves  by  attaching  at  the  reflecting  end  tubes 
of  various  lengths,  which  are  put  in  communication  with  mano- 
metric  flames. 


XV.     DOPPLER'S   PRINCIPLE. 

When  a  source  of  sound  has  a  motion  of  translation  an 
observer  who  listens  will  hear  the  sound  gradually  changing  in 
pitch.  This  may  very  often  be  noticed  in  the  case  of  a  railway 
locomotive,  which  whistles  as  it  is  coming  into  a  station  :  to  an 
observer  at  the  station  the  pitch  of  the  note  seems  gradually  to 
rise. 

If  V  be  the  velocity  of  propagation,  v  the  velocity  of  trans- 
lation, and  \,  \'  the  wave  lengths  of  the  two  notes,  then 

X'_  V±v 
\~      V 

according  as  v  is  in  the  same  or  opposite  direction.  One  of  the 
simplest  ways  to  study  this  is  by  means  of  two  tuning  forks  of 
rather  high  pitch  (512  and  520  v.s.),  which  give  four  beats  per 
second  when  sounded  together  ;  if  an  observer  stations  himself 


136 


EXPERIMENTAL    PHYSICS. 


at  a  distance  of  thirty  or  forty  feet,  and  an  assistant  carries  one 
fork  towards  him,  it  will  be  found  that  the  number  of  beats 
alters ;  if  the  higher  fork  be  carried  at  the  rate  of  about  two 
feet  per  second,  then  nearly  five  beats  a  second  will  be  heard ; 
if  the  lower  fork,  then  about  three. 

This  method  might  be  made  accurate  by 
having  the  tuning  fork  slide  along,  at  a 
uniform  rate,  a  board  twenty  or  thirty  feet 
long,  by  means  of  a  pulley  and  weights, 
which  could  be  adjusted  to  give  the  proper 
speed. 

The  experiment  can  also  be  easily  tried 
by  rotating  a  tuning  fork,  or  by  placing  it 
on  the  slide  of  a  slowly  moving  horizontal 
engine. 

The  apparatus  of  Mac/i,  to  illustrate  this 
principle,  consists  of  a  hollow  brass  tube 
with  reed  attachment  at  one  end  or  at  both 
ends.  This  can  be  rotated  while  the  reed  is 
vibrated  through  a  side  tube,  as  shewn  in 
Fig-  33>  ar)d  an  observer  stationed  in  front 
hears  only  one  note,  while  in  the  plane  of 
Fl?'  33'  rotation  he  hears  the  alternate  increase  and 

diminution   in  pitch.     The  instrument  in  this   state,  however, 
hardly  admits  of  giving  measurements. 


HEAT. 


LIST   OF   EXPERIMENTS. 

I.    Determination  of  the   Zero  and   Boiling-point  of  a  Ther- 
mometer        .........         139 

II.    Calibration  of  a  Tube  ........         140 

III.  Coefficient  of  Expansion  of  Mercury    .....         142 

IV.  Weight  Thermometer.     Cubical  Expansion  of  a  Glass  En- 

velope   .         .         .         .         .         .         .         .         .         .  143 

V.  Coefficient  of  Expansion  of  Dry  Air  at  Constant  Volume  .  146 

VI.  Coefficient  of  Expansion  of  Dry  Air  at  Constant  Pressure  .  149 

VII.  Coefficient  of  Expansion  of  Dry  Air.  —  Third  Method         .  151 

VIII.  The  Air  Thermometer 153 

IX.  Favre  and  Silbermann's  Calorimeter    .         .         .         .         .  155 

X.  Latent  Heat  of  Steam.  —  Regnault's  Method       .         .         .  156 

XI.  Latent  Heat  of  Steam.  —  Berthelot's  Method      .  .159 

XII.  Latent  Heat  of  Vaporization  of  Volatile  Liquids          .         .  160 

XIII.  Weight  of  a  Liter  of  Dry  Air  at  Normal  Pressure  and  Tem- 

perature .........         161 

XIV.  Determination  of  the  Hygrometric  State  of  the  Air    .         .         164 

(0)  The  Chemical  Hygrometer. 
(£)  Condensation  Hygrometers. 

(1)  Regnault's. 

(2)  Alluard's. 

(c}   Regnault's  Experiment  on  Saturated  Vapour  of  Water. 
137 


138  EXPERIMENTAL   PHYSICS. 

XV.    Density  of  a  Vapour.  —  Dumas'  Method      .         .         .         .  1 70 

XVI.   Coefficient  of  Expansion  of  Metals 173 

XVII.    Specific  Heat  of  Dry  Air  at  Constant  Pressure  .        .        .  174 

XVIII.   Pressure  of  Vapours  for  Low  Temperatures         .         .         .  176 

(1)  Between  Zero  and  100°  C. 

(2)  Below  Zero. 

XIX.    Regnault's  Apparatus  for  the  Determination  of  the   Pres- 
sure of  Vapours  at  High  Temperatures        .         .         .  178 


EXPERIMENTS. 


I.     DETERMINATION   OF   THE   ZERO  AND   BOILING-POINT  OF 
A   THERMOMETER. 

To  find  the  zero  point  of  a  thermometer,  any  small  cylindri- 
cal vessel  is  filled  with  pieces  of  clean  ice,  and  the  thermometer 
inserted  so  that  the  zero  marked  on  the  stem  is  just  visible 
over  the  ice.  Holes  made  in  the  bottom  of  the  vessel  allow 
the  water  to  drain  off,  and  the  thermometer  is  left  in  until  the 
column  of  mercury  becomes  stationary.  This  takes  about  half 
an  hour.  The  position  of  the  column  is  then  noted  and  gives 
the  error  of  the  zero  marked  on  the  stem. 

Figure  i  shews  the  arrangement  for  determining  the  boiling- 
point  at  a  given  pressure.  The  thermometer  is  placed  in  a 
vessel  in  which  water  is  boiling,  but  so 
that  the  bulb  does  not  touch  the  water. 
The  steam  circulates  around  it,  and 
passes,  as  shewn  by  the  arrows,  through 
an  outer  jacket  into  the  air,  thus  insur- 
ing that  the  inner  temperature  will  be, 
that  of  the  steam.  A  small  water  mano- 
meter may  be  placed  at  M  to  shew  any 
small  difference  in  pressure  between  the 
steam  inside  and  the  air  outside.  When 
the  mercurial  column  has  become  station- 


Figr.    1. 


ary,  its  position  is  noted  and  at  the  same  time  the  barometer 
is  read  and  reduced  to  zero.  The  true  temperature  of  the 
steam  corresponding  to  the  reduced  height  is  obtained  from 
the  Tables,  and  the  difference  between  this  and  the  observed 

139 


140  EXPERIMENTAL    PHYSICS. 

reading  on  the  stem  gives  the  error  of  the  boiling-point  at  the 
pressure  noted. 

Thus  the  true  temperature,  corresponding  to  any  reading  of 
the  thermometer,  can  easily  be  found. 

In  most  thermometers  sent  out  by  reliable  makers,  the  zero 
point  is  generally  correct  within  a  tenth  of  a  degree,  though 
changing  very  slightly  from  time  to  time;  while  the  boiling- 
point  may  be  in  error  as  much  as  half  a  degree,  and  changes 
every  time  the  thermometer  is  heated  up  to  or  above  100°  C. 

In  connection  with  the  reading  of  temperatures  it  may  be 
noticed  that,  in  a  great  many  experiments  where  the  thermome- 
ter is  used,  it  is  a  difference  of  temperature  that  is  required  : 
in  this  case,  an  ordinary  thermometer  may  be  used  throughout 
the  experiment,  as  its  errors  to  a  large  extent  disappear.  This 
is  notably  the  case  in  finding  specific  and  latent  heats.  And 
when  two  or  more  thermometers  are  used  for  different  measure- 
ments in  the  same  experiment,  it  is  evident  they  should  be  com- 
pared with  one  another  before  and  after  the  experiment. 


II.     CALIBRATION   OF   A   TUBE. 

To  examine  the  bore  of  a  thermometer,  the  apparatus  shewn 
in  the  adjoining  figure  may  be  used. 

A  thread  of  mercury  of  any  desired  length  is  detached  from 


Fig.  2. 


the  column,  and  the  spaces  it  occupies  at  various  points  in  the 
tube  noted.     Usually,  however,  before  the  thermometer  is  made, 


CALIBRATION   OF   A  TUBE.  141 

a  number  of  capillary  tubes  are  examined  by  means  of  this 
instrument,  and  only  a  tube  of  uniform  bore  chosen  for  the 
manufacture  of  the  thermometer. 

To  examine  the  bore  of  a  capillary  tube,  open  at  both  ends, 
a  small  thread  of  mercury  is  introduced  by  attaching  a  piece  of 
rubber  tubing  K  to  one  end,  and  either  allowing  the  mercury  to 
run  in,  or  else  to  be  drawn  up  by  suction. 

The  tube  is  then  placed  under  the  microscopes  M,  N,  which 
are  supported  on  two  movable  stands  C,  D,  alongside  of  a  scale 
graduated  into  millimeters  ;  and  the  lengths  occupied  by  the 
thread  in  different  positions  are  then  measured,  by  means  of 
this  scale,  and  the  micrometer  glasses  in  M  and  N. 

A  curve  may  be  constructed,  of  which  the  ordinates  represent 
the  lengths  occupied  by  the  same  thread,  and  the  abscissae  are 
the  distances  of  one  end  of  the  thread  from  some  fixed  mark 
on  the  tube,  which  may  be  taken  as  the  origin  of  coordinates. 

Most  fine  tubes  will  be  found  slightly  conical  in  bore. 

A  tube  of  rather  large  bore  is  calibrated  by  simply  pouring 
into  it  equal  volumes  of  mercury  determined  by  weighing,  and 
making  corresponding  marks  on  the  tube. 

In  many  cases,  one  may  wish  to  calibrate  a  large  tube  which 
has  already  divisions  on  it  indicating  equal  distances.  The 
readiest  and  most  accurate  way  to  do  this  is  to  fill  the  tube  with 
mercury  to  a  known  mark ;  and  then,  allowing  the  mercury  to 
flow  out  for  a  small  number  of  divisions,  weigh  this  quantity ; 
again  allow  the  mercury  to  flow  out  for  a  small  number  of  divis- 
ions (not  necessarily  the  same  as  before),  weigh,  and  repeat  the 
operation  as  far  as  is  desired.  Then  a  curve  can  be  constructed 
which  will  enable  one  to  read  off  by  interpolation  the  volume 
corresponding  to  any  division  ;  the  fixed  mark  being  the  origin 
of  co-ordinates,  the  numbers  of  divisions  the  abscissae,  and  the 
volumes  the  ordinates.  If  necessary,  the  fixed  mark  may  be 
chosen  as  one  end  of  the  tube,  which  will  then  be  completely 
filled  with  mercury  at  the  first  operation. 


I42 


EXPERIMENTAL    PHYSICS. 


III.    COEFFICIENT   OF   EXPANSION   OF   MERCURY. 

The  most  important  constant,  and  one  upon  which  nearly  all 
others  are  ultimately  based,  in  experiments  with  heat,  is  the 
coefficient  of  expansion  of  mercury.  By  this  is  meant  the 
fraction  which  represents  the  volumetric  increase  in  a  quantity 
of  mercury  for  each  increase  of  temperature  of  i°  C. 

The  method  used  is  that  based  upon  a  principle  suggested  by 
Boyle,  and  devised  by  Dulong  and  Petit.  If  a  bent  tube,  such 

as  is  shewn  in  Fig.  3,  con- 

_  a  z? 

tain  in  its  two  arms  fluids 
of  different  densities,  the 
weight  of  a  cylindrical  col- 
umn of  fluid  in  one  arm 
standing  upon  any  base 
is  equal  to  the  weight  of 
a  cylindrical  column  in  the 
other  arm  standing  on  an 
equal  base  in  the  same 
horizontal  plane. 
c  J)  By  filling  the  tube  with 

Fi£-  3-  mercury,  and  surrounding 

A  C  and  BD  with  envelopes 

so  that  AC  may  be  cooled  to  zero  and  BD  heated  to  the  temper- 
ature of  steam,   we  obviously  can   calculate  the   coefficient  of 
expansion  by  observing  the  heights  of  HQ  and  Ht  from  the  com- 
mon level  below.     For,  conceive  two  small  cylindrical  columns 
standing  on  equal  bases,  whose  heights  from  CD  are  measured. 
The  mass  of  these  two    columns  being  the  same,  since  their 
weights  are  the  same,  their  volumes  are  simply  to  one  another 
as  their  heights,  since  they  stand  upon  equal  bases. 
Let    HQ=  height  of  cold  column. 
Ht=  height  of  hot  column. 
VQ=  volume  of  small  cylindrical  column  in  AC. 
Vt=  volume  of  small  cylindrical  column  in  BD. 
f=  difference  in  temperature. 


WEIGHT   THERMOMETER.  143 

Then 


HQ_Ht 


V»     VI 
HQ_        Vn 


Z!  rtfi+JB)' 


since  the  relation  F=  VQ(i+Kt]  determines  the  coefficient  of 
expansion  K. 

H,  -ffa 


The  apparatus  used  commonly  for  the  determination  of  this 
constant  is  made  by  surrounding  AC  and  BD  with  two  brass 
jackets,  one  of  which  holds  ice,  and  the  other  has  a  tube  leading 
into  it  which  supplies  steam  from  an  adjoining  boiler.  The 
measurements  must  not  be  taken  until  the  temperature  of  the 
hot  column,  as  indicated  by  a  very  accurate  thermometer,  is 
steady.  The  ice  must  be  frequently  renewed,  as  it  melts 
rapidly. 

The  greatest  difficulty  in  the  ordinary  form  of  the  apparatus 
is  to  determine  accurately  the  mean  temperature  of  the  heated 
column.  Measurements  of  heights  must  be  made  as  exact  as 
possible ;  a  small  error  means  a  great  deal  when  the  experiment 
is  performed  with  ice  and  steam,  the  difference  in  level  being 
in  that  case  not  quite  a  centimeter.  Hot  air  or  boiling  oil  may 
be  used  instead  of  steam  ;  but  the  latter  is  exceedingly  dirty 
and  gives  off  disagreeable  odours :  it  has  the  advantage,  how- 
ever, of  giving  a  greater  range  of  temperature. 


IV.     WEIGHT    THERMOMETER:     CUBICAL    EXPANSION    OF    A 
GLASS   ENVELOPE. 

The  absolute  expansion  of  mercury  being  assumed,  the  volu- 
metric expansion  of  a  glass  envelope  can  easily  be  determined 
by  filling  it  with  mercury  at  zero,  and  then  raising  it  to  some 
known  temperature,  when  a  certain  quantity  of  mercury  es- 


144  EXPERIMENTAL   PHYSICS. 

capes,  representing  the  expansion  of  the  enclosed  mercury  less 
the  expansion  of  the  envelope. 

Theory.  — 

Let   WQ  =  weight  of  mercury  filling  the  envelope  at  zero. 
Wt= weight  of  mercury  filling  the  envelope  at  temperature  t. 

k= coefficient  of  expansion  of  mercury. 

S= coefficient  of  expansion  of  glass. 
Then,  weights  being  proportional  to  volumes,  we  have 


For,  the  left-hand  side  of  this  relation  represents  the  internal 
capacity  of  the  envelope  at  temperature  /;  and  Wt  represents 
a  quantity  of  mercury  such  that,  if  allowed  to  expand  from  zero 
to  /,  it  would  just  fill  the  envelope  at  that  temperature. 

From  this  &  is  found,  all  the  other  quantities  being  known. 

Experiment.  —  To  perform  the  experiment,  any  arrangement 
will  suffice  whereby  a  glass  vessel  can  be  completely  filled  with 

mercury  at  zero,  and  then 
heated  to  some  known 
temperature ;  but  the  fol- 
lowing apparatus  will  be 
found  very  convenient. 

A  glass  envelope,  with 
a  capillary  tube  attached 
and  bent  twice,  is  placed 
in  a  receptacle  as  shewn  in 
the  figure  below. 

The  end  of  the  capillary 
tube  dips  into  a  small  iron 
vessel  containing  mercury, 
and  the  bulb  is  prevented 
from  breakage  by  being 
Fig.  4.  surrounded  with  a  small 


WEIGHT   THERMOMETER.  145 

iron  basket  ;  the  whole  being  supported  in  a  circular  brass  plate 
with  a  handle  attached.  The  outer  vessel  is  of  thick  copper, 
and,  when  heated  from  below  with  a  gas-burner,  the  air  inside 
the  bulb  expands  and  bubbles  up  through  the  mercury  in  the 
iron  cup.  Then  it  is  cooled  by  lifting  out  bodily  the  plate  by 
the  handle,  and  exposing  to  the  cool  air.  This  causes  the  air 
in  the  bulb  to  contract,  and  some  mercury  is  then  drawn  in  ; 
and,  by  alternately  heating  and  cooling,  the  bulb  is  finally  filled 
with  mercury.  If  one  wishes  to  perform  the  experiment  more 
rapidly,  the  outer  copper  vessel  may  be  removed,  and  the  basket 
holding  the  bulb  heated  directly  with  a  spirit  lamp  :  this,  how- 
ever, must  be  done  with  care,  as  there  is  danger  of  breakage. 

Then  it  is  surrounded  with  ice  and  cooled  to  zero  ;  and, 
finally,  water  being  put  in  the  copper  vessel,  it  is  heated  to  the 
temperature  of  steam. 

Three  weighings  are  necessary  : 

1.  Weight  of  the  envelope  when  empty  and  perfectly  clean 
and  dry. 

2.  Weight  of  the  envelope  after  heating  to  the  temperature 
of  steam  and  allowing  to  cool  properly. 

3.  Weight  of  mercury  which  escapes  between  zero  and  tem- 
perature /. 

These  weighings  give  WQ  and  Wt. 

Precautions.  —  Heat  slowly,  and  boil  the  mercury  in  the  bulb 
to  remove  air  and  moisture.  Cool  slowly  to  avoid  breakage. 
The  experiment  throughout  must  be  performed  slowly  and 
carefully  to  insure  success. 

It  is  evident  that  the  envelope,  when  once  filled  with  mer- 
cury, and  its  elements  determined,  may  serve  as  a  weight  ther- 
mometer. For,  to  find  the  temperature  at  any  point  where  it 
may  be  placed,  it  is  sufficient  to  fill  it  at  zero  and  then  bring 
it  to  the  place  in  question,  and  notice  how  much  mercury 
escapes.  Then,  as  before, 


and  the  quantities  WQ,   IV,,  S,  k  being  known,  t  is  found. 


146  EXPERIMENTAL   PHYSICS. 

The  weight  thermometer  is  theoretically  a  perfect  instru- 
ment ;  for,  by  its  use,  we  determine  temperatures  in  terms  of 
masses,  which  can  always  be  measured  with  great  accuracy ;  the 
great  objection  to  its  general  use  is  the  length  and  tediousness 
of  the  process. 


V.     COEFFICIENT   OF   EXPANSION   OF   DRY   AIR   AT   CON- 
STANT  VOLUME. 

In  this  experiment  dry  air  is  taken  as  the  type  of  a  perfect 
gas,  and  its  expansion  determined  from  the  increase  of  pressure 
necessary  to  keep  its  volume  constant  when  its  temperature  is 
raised. 

The  apparatus  used  is  that  devised  by  Regnault,  and  is  repre- 
sented in  Fig.  5. 

The  air  to  be  operated  upon  is  inclosed  in  a  glass  bulb  A, 
which  is  placed  in  a  boiler  clamped  to  a  stand  at  K.  The  water 
can  be  run  off  from  this  boiler  by  a  small  stopcock  below,  and 
replaced,  when  necessary,  by  ice,  without  disturbing  the  bulb. 
A  special  perforated  jacket  of  tin  surrounds  the  bulb  inside  to 
insure  a  constant  temperature  when  steam  is  generated.  A 
capillary  tube  of  copper,  attached  to  the  bulb,  is  connected 
at  m,  n  by  means  of  a  three-way  stopcock,  with  a  drying 
apparatus  H,  and  an  exhausting  pump  P,  and  also  with  a 
manometer  of  mercury,  which  is  itself  provided  with  a  three- 
way  tap  at  the  bottom  of  the  tube  BC.  These  three-way  taps 
have  openings  in  the  form  of  a  T,  and  thereby  enable  one  to 
make  communication  between  any  two  of  three  sources,  or  to  shut 
any  two  off,  or  all  three ;  the  directions  in  which  the  openings 
run  are  shewn  on  the  outside  by  small  lines.  The  drying  tubes 
contain  pumicestone  moistened  with  sulphuric  acid. 

The  connections  are  all  made  so  that  no  leakage  takes  place, 
and  the  manometer  tubes  are  filled  with  mercury  until  they  both 
stand  at  the  same  level  a,  everything  being  in  communication 
with  the  air.  Then  the  three-way  tap  at  the  bottom  of  C  is 


COEFFICIENT    OF    EXPANSION    OF   DRY   AIR.  147 

closed,  so  that  the  tube  BC  does  not  communicate  with  DE. 
The  process  of  drying  the  air  then  commences.  Water  is 
placed  in  the  boiler  and  steam  generated,  and  then  exhaustion 
is  made  by  P,  through  the  drying  tubes  ;  finally  air  is  allowed 
to  enter  by  means  of  a  central  tap  placed  at  the  bottom  of  P 
which  makes  direct  connection  with  the  air.  This  operation  is 


Fig.   5. 

repeated  a  number  of  times,  the  water  still  boiling,  and,  at 
length,  when  the  air  is  dry,  the  tap  at  C  is  opened,  and  the 
apparatus  throughout  brought  to  atmospheric  pressure.  Then 
the  hot  water  in  the  boiler  is  run  off,  replaced  by  ice,  and  the 
air  in  A  cooled  to  zero ;  and  when  the  ice  has  been  in  about 
twenty  minutes,  the  stopcock  at  n  is  turned  so  that  A  com- 


148  EXPERIMENTAL   PHYSICS. 

municates  only  with  the  manometer.  The  barometer  is  then 
read,  and  the  levels  of  the  two  columns  of  mercury  in  the 
manometer  (which  should  be  the  same)  noted. 

There  is  now  a  volume  of  dry  air  inclosed  at  zero,  and  at  a 
known  pressure.  The  next  operation  is  to  heat  this  air  to  the 
temperature  of  steam,  and  as  it  expands,  driving  before  it  the 
column  aC,  to  pour  mercury  in  ED  so  that  it  keeps  the  air 
always  at  constant  volume,  and  the  mercury  in  BC  therefore 
always  at  the  mark  a.  This  second  operation  will  last  about 
fifteen  minutes  ;  and  when  the  column  in  DE  remains  station- 
ary the  difference  in  level  in  the  two  tubes  is  noted  and  the 
barometer  again  read. 

Theory.  — 

Let  V§  =  volume  of  air  inclosed  in  A  at  zero. 
7'  =  volume  between  the  marks  ;/  and  a. 
HQ  =  barometric  reading  when  air  is  at  zero. 
H  —  barometric  reading  when  air  is  heated. 
//  =  difference  in  level  in  the  two  tubes  at  the  end  of  the 

experiment. 

T=  temperature  of  steam  when  h  is  read. 
/=  temperature  of  room. 

8  =  coefficient  of  expansion  of  the  glass  envelope. 
«  =  coefficient  of  expansion  of  dry  air. 

Then,  by  Boyle's  law, 


i+a/  l+aT          l+ 

where  VQ(i  +  BT)  represents  the  internal  capacity  of  the  glass 
envelope  at  the  temperature  7!  The  coefficient  of  expansion  of 
glass  may  be  obtained  from  the  tables  ;  but  this  is  only  approxi- 
mate, as  its  value  depends  more  on  the  form  of  the  envelope  than 
on  the  quality  of  the  glass,  and  for  accurate  purposes  should  be 


COEFFICIENT   OF   EXPANSION    OF  DRY    AIR.  149 

determined  in  each  case.  The  capacity  of  the  bulb  at  zero  may 
be  found  in  the  usual  way  and  marked  on  the  outside. 

In  the  preceding  formula  a  first  approximation  to  the  value  of 
a  is  found  by  neglecting  v.  This  value  is  then  used  to  reduce  the 

expression  — - —  in    the  second    approximation.     The  experi- 
i  +  at 

ment  may  also  be  performed  in  the  reverse  order,  by  filling  the 
bulb  with  hot  air  at  atmospheric  pressure  and  then  cooling  it  to 
zero  :  in  this  case  «  represents  a  contraction,  and  the  formula 
will  be  altered  correspondingly. 

Precautions.  —  In  surrounding  the  bulb  with  ice,  after  it  has 
been  raised  to  the  temperature  of  steam,  care  must  be  taken 
not  to  place  ice  on  the  metal  socket  of  the  glass  too  suddenly. 
It  is  better  to  run  off  the  hot  water,  then  cool  the  metal  boiler 
with  cold  water,  and  finally  the  bulb.  The  stopcock  at  C  must 
be  closed  when  drying  the  air,  otherwise  the  mercury  will  run 
over  into  the  drying  tubes  and  bulb.  When  finished,  the  appa- 
ratus should  be  opened  to  the  air  and  the  mercury  in  the  tubes 
let  down  to  the  same  level.  The  use  of  the  three-way  tap  at  C 
and  at  n  must  be  fully  understood. 


VI.    COEFFICIENT  OF  EXPANSION  OF  DRY  AIR  AT  CON- 
STANT PRESSURE. 

The  mode  of  operation  in  this  experiment  is  similar  to  that  in 
the  previous  one,  the  only  change  being  that  when  the  dry  air 
is  heated  from  zero  to  the  temperature  of  steam  the  mercury  is 
allowed  to  flow  out  until  the  two  columns  are  at  the  same  level. 
The  preceding  formula  then  applies,  h  being  zero,  and  v  on  the 
left-hand  side  becoming  v'  on  the  right-hand  side. 

The  apparatus  used  is  shown  in  Fig.  6. 

The  tubes  BC,  ED  are  inclosed  in  a  water  jacket  to  insure 
constancy  of  temperature,  the  bulb  A  is  placed  as  before  in  a 
boiler  (not  shewn),  and  the  connections  at  m  and  n  lead  by  a 
three-way  tap  to  drying  tubes  and  an  air-pump. 


150  EXPERIMENTAL   PHYSICS. 

FJ 


Theory.  —  Fie-  6- 

Let  VQ  =  volume  of  bulb  at  zero. 

v  =  small  volume  to  a  chosen  mark. 

v'  —  volume  when  BC  is  lowered. 
HQ=  barometric  pressure  at  zero. 
H '  =  barometric  pressure  at  temperature  of  steam. 

T=  temperature  of  steam. 

/  =  temperature  of  water. 

g  =  coefficient  of  expansion  of  glass. 

a  =  coefficient  of  expansion  of  air  at  constant  pressure. 


COEFFICIENT   OF   EXPANSION   OF  DRY   AIR. 


Then,  by  Boyle's  law, 
•   .       v 


~^H. 


I  +  at 

In  this  v'  will  be  much  greater  than  v\  both  are  found  by  filling 
the  tube  BC  with  mercury  to  the  tap  n,  and  then  allowing  it  to 
run  out  to  the  marks  which  define  v  and  v' ;  the  volumes  are 
then  inferred  from  the  weights  of  the  quantities  of  mercury 
which  escape.  First  and  second  approximations  are  made  as 
before,  and  the  general  precautions  are  the  same  as  in  the 
former  experiment. 


VII.    COEFFICIENT    OF    EXPANSION    OF    DRY   AIR.  — THIRD 
METHOD. 

In  this  experiment  dry  air  is  cooled  from  the  temperature  of 
steam  to  zero,  and  both  volume  and  pressure  change.  A  small 
cylindrical  bulb  of  glass  with 
a  bent  capillary  stem  is  used, 
and  filled  with  dry  air  by  being 
placed  in  the  apparatus  for 
determining  the  boiling-point 
of  a  thermometer  (see  Fig.  i), 
in  connection  with  a  drying 
tube  and  an  exhausting  air- 
pump.  When  filled  with  dry 
air  by  continual  exhaustion  and 
heating,  the  end  is  sealed  up, 
after  detaching  the  drying- 
tube  a  few  seconds  to  allow  it 
to  come  to  atmospheric  pres- 
sure. It  is  then  placed,  when 
cool,  in  the  apparatus  below 
(Fig.  7). 

AB  is  the  glass  envelope  inverted  in  a  trough  of  mercury  and 
surrounded  with  melting  ice  placed  in  a  vessel  shewn  in  section 


152  EXPERIMENTAL    PHYSICS. 

by  EE.  D  is  a  small  outlet  for  the  melted  ice.  At  C  is  an 
arm  which  can  be  moved  up  and  down,  and  to  this  is  attached  a 
small  pipe-shaped  piece  of  metal  whose  bowl  is  filled  with  wax ; 
this  can  be  pushed  in  against  the  end  of  the  capillary  tube,  and 
enables  one  to  close  it  under  the  mercury.  H  is  a  screw  which 
can  be  adjusted  to  touch  the  surface  of  the  mercury. 

Order  of  the  Experiment.  —  i.  Determine  the  coefficient  of 
expansion  of  the  glass  bulb  in  the  same  manner  as  given  in 
Exp.  4. 

2.  Place  it  in  the  apparatus  shewn  in  Fig.   i,  and   connect 
with  drying-tubes  and  an  air-pump ;   and  when  the   air   in    it 
is  completely  dry,  allow  it  to  come  to  atmospheric  pressure  by 
separating  it   from  the   drying-tubes   (while  the  steam  is  still 
circulating  around  it)  for  a  few  seconds.     Then  seal  the  end 
with    a   blow-pipe,    at    the    same   time    noting   the    barometric 
pressure. 

3.  Arrange  as  in  Fig.  7,  and  break  off  the  point  underneath 
the  mercury.     This  is  done  most  readily  by  making  a  small  file 
mark  near  the  end  before  inverting  it,  and  then  with  a  pair  of 
pliers  breaking  it  off  under  the  mercury,  at  the  same  time  hold- 
ing firmly  the  stem  at  B. 

The  mercury  then  rises  into  the  capillary  tube,  and  on  filling 
EE  with  ice,  it  finally  takes  up  a  permanent  position.  Then 
the  screw  H  is  moved  down  until  its  lower  point  touches  the 
surface  of  the  mercury,  and  the  bowl  of  soft  wax  is  pressed  in 
against  the  end  of  the  tube,  sealing  it  up,  and  at  the  same  instant 
the  barometer  is  read.  The  ice  being  next  carefully  removed, 
and  everything  allowed  to  come  to  the  temperature  of  the  room, 
the  height  from  the  top  of  //"to  the  top  of  the  column  of  mer- 
cury is  measured  with  a  cathetometer,  and  then,  finally,  the 
length  of  the  screw  H.  The  bulb,  is  then  removed,  and  the 
mercury  taken  from  it  and  weighed. 

Theory.  — 

Let  VQ  =  volume  of  mercury  filling  the  bulb  at  zero. 
8=  coefficient  of  expansion  of  the  glass  bulb. 


THE    AIR   THERMOMETER.  153 

HQ  =  barometric  reading  when  the  end  is  sealed  up  under 

the  mercury. 
H=  barometric  reading  when  sealed  at  the  temperature 

of  steam. 

T=  temperature  of  steam. 
h  =  height  of  column. 

v= volume  of  mercury  which  runs  into  bulb  when  cooled. 
a  =  coefficient  of  expansion  of  dry  air. 

Then,  by  Boyle's  law, 

(  r0-v)(*f0-/i)(i  +aT)=  F0(i  +&T)H, 

from  which  a  is  determined. 

Precautions.  —  Care  must  be  taken  in  determining  VQ  and  8. 
Some  special  apparatus  should  be  used  for  filling  the  bulb  with 
mercury,  either  by  the  application  of  heat  or  by  means  of  a 
good  air-pump  and  a  three-way  tap.  The  air  must  be  well 
dried.  In  finding  v  by  weighing  the  mercury,  some  difficulty 
will  be  found  in  getting  it  out  of  the  bulb ;  it  is  well  to  weigh 
the  bulb  previous  to  the  experiment,  and  then  weigh  bulb  and 
mercury  at  the  end  of  the  experiment,  checking,  if  possible,  by 
weighing  separately  the  mercury.  A  small  error  is  introduced 
by  the  wax  and  the  piece  broken  off ;  but  these  can  be  allowed 
for  without  any  difficulty.  The  values  of  //0,  H,  h  are  sup- 
posed to  be  taken  for  some  standard  temperature,  usually  zero. 

VIII.     THE  AIR   THERMOMETER. 

The  principle  of  the  air  thermometer  is  that  a  perfect  gas, 
such  as  dry  air,  if  kept  at  constant  volume,  will  have  its  pressure 
increased  as  its  temperature  increases  :  the  constant  of  increase 
being  0.003665  for  each  degree  centigrade.  Any  of  the  preced- 
ing instruments  (Figs.  5,  6,  7)  may  be  used  as  an  air  thermome- 
ter, assuming  the  value  of  «.  The  most  convenient  apparatus 
is  one  similiar  to  that  of  Exp.  5.  The  air  thermometer  itself 
is  a  glass  bulb,  or  (for  very  high  temperatures)  a  cylindrical 


154  EXPERIMENTAL   PHYSICS. 

or  spherical  reservoir  of  Bayeux  porcelain,  with  a  long  capillary 
tube  of  copper,  so  that  it  may  be  carried,  if  necessary,  some 
distance  from  the  apparatus.  This  thermometer  is  connected 
to  a  manometer,  drying  apparatus,  and  air-pump,  just  as  in 
the  former  experiments.  Then,  having  obtained  dry  air,  it  is 
cooled  to  zero,  and  the  barometer  read  :  the  two  columns  of  the 
manometer  being  at  the  same  level.  Then  the  stopcock  lead- 
ing to  the  air-pump  is  closed,  and  the  bulb  is  placed  where  the 
temperature  is  desired,  and  as  the  air  in  it  expands,  the  volume 
is  kept  constant  and  the  increase  of  pressure  and  barometric 
reading  taken.  Then  the  formula  of  Exp.  5  applies  : 

} 

V 


I  +aT       I  +a/ 

where  a  is  now  assumed,  and  T,  the  unknown  temperature,  is 
to  be  found.  This  process  can  be  used  to  graduate  a  mercurial 
thermometer  :  the  reservoir  of  air  is  raised  from  zero  to  tempera- 
tures gradually  increasing,  and  the  temperatures  calculated 
from  the  formula  are  compared  with  the  indications  of  the 
mercurial  thermometer. 

For  rough  calculations  v  may  be  neglected,  when  it  is  very 
small,  and  also  8  ;  then  it  is  not  necessary  to  know  F"0,  and  we 
get  the  temperature  from  the  simple  relation 


and 


0.003665 


The  essential  precaution  in  the  use  of  the  air  thermometer 
is  to  have  the  air  perfectly  dry:  this  is  insured  by  the  use 
of  a  good  Sprengel  or  Geissler  pump  in  connection  with  a 
reservoir  of  sulphuric  acid. 

The  porcelain  reservoir  is  used  for  the  determination  of 
temperatures  beyond  that  at  which  glass  begins  to  melt. 


FAVRE   AND   SILBERMANN'S   CALORIMETER. 


155 


IX.     FAVRE   AND   SILBERMANN'S   CALORIMETER. 

This  is  in  reality  a  large  thermometer,  consisting  of  a  reservoir 
A  (Fig.  8),  which  contains  about  fifty  pounds  of  mercury  and 
which  has  a  graduated  stem  CD  with  a  fine  bore  into  which  the 
mercury  of  the  reservoir  when  heated  can  expand. 

A  microscope  L  provided  with  a  micrometer  glass  enables  one 
to  estimate  the  divisions  on  CD  to  twentieths  of  a  millimeter. 
A  special  contrivance  at  B,  with  a  handle,  is  used  to  put  pressure 


Fig.  8. 


on  the  interior  of  the  reservoir  and  move  the  thread  of  mercury 
to  any  desired  point ;  and  there  are  two  or  three  openings  in  A 
in  which  heated  bodies  may  be  placed  and  thereby  give  out 
their  heat  to  the  mercury  in  the  reservoir. 

The  instrument  is  first  graduated  by  taking  a  known  weight 
of  water  at  a  known  temperature,  and  inserting  it,  as  shewn  at 
E,  and  noticing  the  number  of  divisions  over  which  the  thread 
D  moves,  as  the  heat  from  this  water  is  imparted  to  A. 


156  EXPERIMENTAL   PHYSICS. 

The  theory  of  the  instrument  is  simple. 
Let  M=mass  of  water  introduced. 

T=  temperature  when  introduced. 

0=;its  temperature  when  the  thread  is  stationary. 

«  =  number  of  divisions  over  which  the  thread  has  moved. 

Then  — ' — =-^-  =  value  of  one  division  in  units  of  heat 


To  measure,  then,  a  specific  heat  of  a  liquid,  for  example, 
Let  P=mass  of  liquid  introduced. 

T=  its  initial  temperature. 

0'  =  its  final  temperature. 

x=its  specific  heat. 

m  —  number  of  divisions  over  which  the  thread  moves. 

Then  Px(T-6')  =  mc, 

from  which  x  is  found. 

The  instrument  can  be  used  for  measuring  specific  heats  of 
liquids,  heats  of  combustion,  and  even  specific  heats  of  solids. 
Care  only  must  be  taken  in  determining  the  constant  c,  which 
should  be  found  from  the  mean  of  a  number  of  observations. 

X.    LATENT   HEAT   OF   STEAM.  —  REGNAULT'S   METHOD. 

This  is  essentially  a  method  of  mixture,  the  apparatus  for 
performing  the  experiment  being  shewn  in  Fig.  9. 

B  is  a  boiler  in  which  steam  is  generated  either  at  atmos- 
pheric or  other  pressures.  The  steam  is  led  from  this  boiler 
into  a  brass  globe  G  placed  in  a  calorimeter  C  containing  a 
known  weight  of  water.  The  handle  H  when  turned  in  a  cer- 
tain direction  allows  the  steam  to  pass,  by  a  peculiar  arrange- 
ment of  two  hollow  tubes  working  within  one  another,  into  G ; 
when  turned  in  all  other  directions  the  steam  escapes  through 


LATENT   HEAT  OF   STEAM. 


157 


an  exhaust  pipe  E  and  condenses  in  a  vessel  below.  The  pipes 
P,  P  are  used  when  pressures  greater  than  atmospheric  are 
needed ;  they  lead  to  a  brass  box,  which  can  be  connected  with 
a  mercurial  manometer,  or  gauge  of  any  kind  to  indicate  the 
pressure  at  which  the  steam  is  generated.  The  steam  passes 
then  into  G,  condenses,  and  gives  out  heat  to  the  surrounding 

.f 


Fig.  9. 

water  in  the  calorimeter,  which  rises  correspondingly  in  tem- 
perature. After  a  certain  time  has  elapsed,  the  steam  is  shut 
off  by  turning  H,  and  then,  waiting  a  few  minutes  to  allow  the 
condensed  steam  in  G  to  come  to  its  lowest  temperature,  the 
tap  T  is  opened  and  the  water  of  condensation  collected  and 
weighed  or  estimated  by  volume.  The  temperature  of  the 
water  in  the  calorimeter,  at  the  beginning  and  end  of  the 
experiment,  is  taken  with  a  thermometer  reading  directly  to 
fifths  or  tenths. 

Theory.  — 

Let  W= weight  of  water,  together  with  the  water  equivalent 
of  the  calorimeter. 

/=its  temperature. 

w=  weight  of  condensed  steam. 

T=  temperature  of  the  steam. 


158  EXPERIMENTAL   PHYSICS. 

T'=  temperature  of  condensed  steam  when  run  out  from 

the  calorimeter. 

0  =  final  temperature  of  the  water  in  the  calorimeter. 
x=  latent  heat  of  the  steam. 

Then,  by  the  equivalence  of  heat  lost  and  gained, 


For  the  left-hand  side  of  this  relation  represents  the  heat 
absorbed  by  the  water  and  calorimeter;  and  the  right-hand  side 
is  the  heat  given  out  by  the  steam  in  condensing,  together  with 
the  heat  given  out  by  the  condensed  steam  in  falling  from  the 
boiling-point  to  its  final  temperature  when  run  off. 

Precautions.  —  i.  The  greatest  difficulty,  in  such  an  experi- 
ment as  this,  is  to  determine  accurately  the  water  equivalent  of 
the  calorimeter,  which  may  either  be  found  by  an  independent 
experiment,  using  the  method  of  mixture,  or  determined  in 
conjunction  with  the  latent  heat  of  steam  by  the  method  of 
successive  approximation. 

2.  The  water  in  the  calorimeter  should  be  in  the  first  place 
as  cold  as  possible. 

3.  The  steam,  in  passing  into  the  calorimeter,  is  made  to  heat 
the  water  as  much  above  the  temperature  of  the  room  as  it 
previously  was  below.     This  is  to  equalize  radiation  and  absorp- 
tion.    If  the  temperature  of  the  room  be  above  18°  C.,  then  the 
initial  temperature  of  the  water  should  not  be  more  than  8°  C.  ; 
otherwise  the  final  temperature  of  the  water  will  be  so  high  that 
rapid  evaporation  takes  place  from  its  surface. 

4.  A  series  of  temperatures  should  be  taken  as  the  condensed 
steam  is  running  out,  and  the  mean  in  some  way  accurately 
found. 

5.  The  water  should  be  well  stirred  just  before  the  steam  is 
let  into   the  calorimeter,   and    its   temperature   carefully  read. 
Stirring  should  be  kept  up  throughout  the   experiment,  espe- 
cially if  the  surface  of  the  water  is  near  the  pipe  which  enters 


LATENT   HEAT   OF   STEAM. 


159 


into  G,  for  in  that  case  the  surface  water  becomes  very  hot  and 
gives  off  steam. 

6.    Care  should  be  taken  not  to  let  the  boiler  run  dry,  and 
when  H  is  closed,  to  open  the  exhaust  pipe  E,  or  the  outlets  P,  P. 


XI.     LATENT   HEAT   OF   STEAM.  —  BERTHELOT'S   METHOD. 

Although  the  preceding  method  can  be  applied  to  calculate 
the  latent  heat  of  steam  and  other  vapours  at  different  pressures, 
yet  it  necessitates  large  quantities  of  liquid,  and  the  errors  of 
experiment  are  not  easily  corrected.  Berthelot's  arrangement 
is  more  simple,  avoiding  thereby  several  sources  of  error;  it 
can  be  used  for  any  volatile  substance,  requires  but  a  small 
quantity  of  liquid  with  which  to  operate,  and  gives  accurate  and 
rapid  results. 

It  consists  of  a  calorimeter  C  (Fig.  10)  of  special  construc- 
tion, filled  with  water  or  other  non-conducting  material,  and 
covered  with  felt,  and  an  inner 
vessel  c  in  which  a  known 
quantity  of  water  is  placed. 

The  boiler  of  the  preceding 
experiment  is  replaced  by  a 
bottle-shaped  vessel  G,  in 
which  the  liquid  to  be  exam- 
ined is  placed  ;  and  this,  being 
heated  from  below,  sends  va- 
pour, as  shewn  by  the  arrows, 
through  a  tube  AD  into  a 
receptacle  B,  where  it  con- 
denses and  gives  out  its  heat ; 


Fig.  10. 


owing  to  the  compact  form  of  the  apparatus,  no  heat  is  lost  in 
the  passage  from  G  to  B.  At  D  is  a  detachable  clamp  by 
means  of  which  one  can  separate  the  boiler  from  B ;  and  a 
spiral  tube  generally  runs  from  D  to  B,  so  that  condensation  is 
more  easily  effected.  The  exit  at  E  insures  that  vaporization 


i6o 


EXPERIMENTAL   PHYSICS. 


takes  place  at  atmospheric  pressure.  The  boiler  and  receptacle, 
with  attached  tubes,  may  be  made  of  glass  or  metal ;  if  of  glass, 
then  the  instrument  can  be  used  for  almost  any  volatile  liquid, 
and  has  the  advantage  of  being  transparent. 

The  water  equivalent  of  the  calorimeter  in  this  case  is  easily 
found  by  detaching  B  and  weighing  it ;  and  the  heat  from  the 
burner  which  is  placed  under  G  can  be  prevented  from  heating 
the  water  in  c  by  placing  over  the  lid  of  the  apparatus  a  layer 
of  asbestos. 

The  thermometer  /  should  be  graduated  in  tenths  of  degrees. 

To  perform  the  experiment,  unfasten  the  clamp  D,  weigh  B 
and  connecting  tubes,  and  determine  its  water  equivalent ;  then 
invert  G,  pour  in  the  liquid  to  be  heated,  and  arrange  as  in  the 
figure.  The  method  being  one  of  mixture,  the  calculation  is 
the  same  as  in  the  preceding  experiment. 


XII.     LATENT   HEAT   OF   VAPORIZATION   OF   VOLATILE 
LIQUIDS. 

For  liquids  which  evaporate  rapidly  at  ordinary  temperatures 

a  reverse  process  devised  by  Regnault  may  be  used. 

It  consists  in  evaporating  a  quantity  of  the  liquid  at  a  low 

pressure    in    a   vessel    surrounded   with   water,    and    from    the 

diminution  in  tempera- 
ture of  this  water  cal- 
culating the  heat  nec- 
essary for  vaporization. 
The  apparatus  used  is 
similar  to  that  shewn  in 
Fig.  ii. 

The  liquid  to  be  evap- 
orated is  poured  into  a 
calorimeter  C  through  a 

tube  at  O,  which  is  then  closed  with  a  tight-fitting  screw  cap. 

The    calorimeter   portion   consists  of   three  vessels :   an  inner 


WEIGHT   OF   A   LITER   OF   DRY   AIR.  161 

closed  one  in  which  the  liquid  is  placed,  a  second  to  hold 
the  water  to  be  cooled,  and  an  outer  one  to  prevent  direct 
radiation  or  absorption  of  heat.  A  tube  leads  from  the  inner- 
most closed  vessel  to  another  small  calorimeter  c,  and  then  to 
an  air  pump  connected  at  A.  The  tube  M  is  connected  with 
a  pressure  gauge  or  manometer. 

A  freezing  mixture  is  generally  placed  in  c,  so  that  most  of 
the  vapour  produced  by  evaporation  in  C  condenses,  and  can 
be  collected  again  by  opening  the  tap  t.  A  stirrer  and  ther- 
mometer are  also  necessary. 

Theory.  — 

Let  W=  weight  of  water  in  the  calorimeter. 
w  =  water  equivalent  of  the  calorimeter. 
T=  initial  temperature  of  the  water. 

t=  final  temperature  of  the  water. 
P  =  weight  of  liquid. 

c  =  its  specific  heat. 

x= latent  heat  of  vaporization. 

6  =  temperature  of  vaporization. 

Then 

(W+w)(T-t)+Pc(T-0)=Px, 

from  which  x  is  found. 

In  addition  to  the  term  Px  there  will  be  another  term  (due  to 
the  vapour  leaving  the  calorimeter  becoming  heated),  which  would 
involve  the  specific  heat  of  the  vapour ;  but  it  will  generally  be 
very  small.  There  is  also  a  slight  uncertainty  in  the  determina- 
tion of  the  temperature  of  ebullition  (0). 

XIII.     WEIGHT   OF   A    LITER   OF    DRY    AIR    AT   NORMAL 
PRESSURE   AND   TEMPERATURE. 

The  method  employed  is  that  of  Regnault,  in  which  two 
hollow  glass  globes  are  blown  as  nearly  as  possible  identical  in 
every  respect,  the  capacity  of  each  being  about  seven  liters.  One 


162 


EXPERIMENTAL   PHYSICS. 


of  these  is  sealed  up  permanently,  and  the  other  is  provided 
with  a  stopcock.  They  are  suspended  by  hooks  from  the  scale 
pans  of  a  large  balance  and  are  brought  into  a  state  of  equi- 
librium by  the  addition,  if  necessary,  of  small  weights  ;  if  a  very 
accurate  experiment  is  required,  they  are  next  suspended  in 
water,  and  due  allowance  made  for  any  difference  in  their  exter- 
nal volume  shewn  by  the  unequal  upward  pressures. 

In  this  way  one  obtains  two  bodies  which  displace  equal 
amounts  of  air,  whatever  be  its  pressure,  temperature,  and 
hygrometric  state. 

The  globe  provided  with  a  stopcock  is  then  removed  and 
placed  in  connection  with  an  air  pump  and  mercurial  gauge, 
and  drying  apparatus,  as  shewn  in  the  adjoining  figure. 

G  is  the  globe,  which  may  be  surrounded  either  with  water 
of  a  known  temperature  or,  better,  with  ice.  T'  is  the  stop- 

\M 


Fig.  12. 

cock,  which  when  open  makes  communication  with  a  three-way 
tap  /'attached  to  a  stand  5.  At  A  is  an  outlet  leading  to  a 
drying  apparatus,  gauge,  and  air  pump. 

When  T'  and  T  are  so  adjusted  that  the  globe  G  communi- 
cates with  A,  it  may  be  filled  with  perfectly  dry  air  at  a  known 
temperature,  usually  zero :  it  is  then  left  in  communication  with 
the  air  for  a  few  seconds  and  the  barometric  reading  taken. 

After  assuming  the  temperature  of  the  room,  it  is  again  put 


WEIGHT   OF   A   LITER   OF   DRY    AIR.  163 

in  the  scale  pan,  and  equilibrium  established  with  the  compen- 
sating globe  and  small  weights.  Once  more  it  is  removed, 
placed  in  connection  with  the  air  pump  and  gauge,  and  exhausted 
as  completely  as  possible  ;  and  the  indication  of  the  gauge  along 
with  the  barometric  reading  taken.  Finally,  after  closing  T't 
it  is  placed  on  the  scale  pan,  and  equilibrium  restored  by  the 
addition  of  certain  new  weights,  which  must  represent  the 
weight  of  the  dry  air  removed  from  the  globe. 

Then  it  only  remains  to  determine  the  internal  capacity  of  G 
at  zero,  which  is  done  by  taking  the  exhausted  globe  and  plac- 
ing it,  by  means  of  T,  in  connection  with  the  outlet  M,  without 
opening  T1.  A  tube  being  now  led  from  M  into  a  vessel  of 
water,  T'  is  opened,  and  the  water  rushes  from  M  into  G  and 
almost  completely  fills  it.  The  opening  in  the  top  of  the  globe 
is  usually  so  small  that  it  would  take  some  time  to  fill  it  in  any 
other  way. 

Theory.  — 

Let  VQ=  internal  capacity  of  the  globe  at  zero. 

//  =  mean  of  the  two  barometric  readings  taken. 
h  =  reading  of  the  gauge  when  the  globe  is  exhausted. 
P  =  weight  necessary  in  the  final  weighing  to  produce 
equilibrium. 

Then,  by  Boyle's  law,  the  volume  of  air  removed  is 
vH-h 

V«~IT' 

and  therefore  the  weight  of  a  liter  of  air  at  o°  C.  and  760  mm.  is 


F0     H-h 

Precaution.  —  Care  must  be  taken,  when  determining  the 
capacity  of  the  glass  globe,  not  to  allow  the  water,  by  any  mis- 
take in  turning  T,  to  get  into  the  air  pump  through  A. 

It  is  evident  that  the  above  method  can  be  readily  applied  to 
the  determination  of  the  densities  of  gases. 


1 64  EXPERIMENTAL   PHYSICS. 

XIV.     DETERMINATION    OF  THE    HYGROMETRIC    STATE 
OF    THE   AIR. 

The  hygrometric  state  of  the  air  will  be  known  when  we 
know  the  quantity  of  water  vapour  in  a  given  volume  of  air, 
and  also  the  pressure  which  this  vapour  exerts ;  but  the  deter- 
mination of  either  of  these,  although  simple  in  theory,  is 
attended  with  great  practical  difficulties. 

The  hygrometric  coefficients  may  be  found  by 

(a)    The  chemical  hygrometer. 

This  consists  simply  in  passing  a  known  volume  of  air 
through  drying  tubes  which  are  weighed  before  and  after  the 
experiment. 

The  weight  of  vapour  in  a  given  quantity  of  air  is  thus  found, 
and  if  we  wish  the  pressure  which  this  vapour  exerts,  we  may 
find  it  in  the  following  way. 

Let  w  =  weight  of  water  in  grains. 
?/=its  volume  in  liters. 
/=its  temperature. 
;r=its  pressure. 

Then,  since  i  liter  of  air  at  o°  C.  and  760  mm.  pressure 
weighs  1.293  grams,  and  since  water  vapour  is  .622  times 
denser  than  air,  we  have 


_vxx  1.293  x  .622 
(1+^)760 


where 
And 


Conversely,  if  we  know  x,  we  can  find  w. 

If  we  require  to  find  the  weight  of  a  given  volume  of  moist 
air, 

Let  V=  volume  in  cc. 

H  =  barometric  pressure. 


HYGROMETRIC   STATE   OF   THE   AIR. 


I65 


f— pressure  of  vapour  of  water  in  it,  found  from  the  dew- 
point  and  the  tables. 
t—  temperature  of  air. 
«  =  . 003665. 

Then  the  weight  of  dry  air  in  V 


Fx.ooi2Q3  x 


And  the  weight  of  vapour 
=  .622  Fx. 001293  x 

Therefore  the  weight  required 
_Fx.ooi293  tZ7 


760 


Fx" °°1293 (#-f/). 
760(1+ «/)  V 

To  perform  the  experiment  (a),  a  set  of  drying  tubes  is 
attached  to  an  aspirator  (shewn  in  section  in  Fig.  13),  consist- 
ing of  two  reservoirs  R,  R, 
capable  of  holding  water, 
and  at  the  same  time  of 
rotating  about  a  fixed 
horizontal  axis  AB.  By 
means  of  the  tubes  C,  D, 
the  water  falling  from  the 
upper  reservoir  into  the 
lower  drags  air  through  the 
opening  at  A  and  forces  it 
through  the  exit  E,  as  in 
the  figure.  It  will  also  be 
seen  that  when  the  lower 
reservoir  is  full,  the  instru- 
ment can  be  inverted,  and 
the  same  process  repeated 
without  disturbing  any  con- 
nections made  at  A  or  E. 


Any  required  volume  of  air 


Fig.  13. 


166  EXPERIMENTAL   PHYSICS. 

can  then  be  operated  upon  in  a  short  time.  The  drying  tubes 
are,  of  course,  connected  at  A. 

The  volume  of  air  dragged  through  the  aspirator  at  one 
operation  may  be  taken  approximately  to  be  the  capacity  of 
the  reservoir,  which  is  supposed  to  be  initially  full  of  water. 
If  a  very  accurate  experiment  is  required,  account  will  have 
to  be  taken  of  the  change  in  pressure  owing  to  the  final  air 
in  the  aspirator  being  saturated  with  water  vapour  and  at  a 
different  temperature  from  that  which  came  to  the  drying 
tubes. 

It  is  preferable  to  have  three  drying  tubes,  two  for  the 
absorption  of  the  moisture  from  the  air  upon  which  one 
operates  ;  and  the  third,  placed  next  the  aspirator,  to  absorb 
any  moisture  which  may  chance  to  get  back  from  the  aspira- 
tor, which  is  filled  with  water :  the  two  outside  ones  are  then 
weighed  before  and  after  the  experiment. 

Tubes  filled  with  calcium  chloride  or  pumice  stone  and 
sulphuric  acid  may  be  used,  and  instead  of  making  them  air- 
tight with  paraffin  or  wax,  it  is  better  that  they  should  be 
provided  with  perforated  glass  stoppers :  they  can  then  be 
opened  or  closed  at  will. 

(b)    Condensation  hygrometers. 

The  determination  of  the  hygrometric  state  of  the  air  by 
chemical  means  is  slow,  and  does  not  give  accurate  informa- 
tion of  the  variations  in  humidity  during  short  intervals  of  time: 
it  is  really  a  mean  of  the  states  which  the  air  had  during  the 
time  of  the  experiment.  More  minute  and  rapid  information 
is  obtained  by  the  hygrometers  of  condensation,  which  are 
arrangements  whereby  the  temperature  inside  a  vessel  is  so 
lowered  that  dew  appears  on  the  outside  owing  to  condensation 
of  moisture  from  air  in  its  vicinity.  A  glass  of  ice  water  on 
a  hot  day  is  the  simplest  form  of  hygrometer ;  and  the  forma- 
tion of  dew  on  the  earth  when  it  loses  its  heat  by  radiation 
into  space  shews  how  the  earth  itself,  under  favourable  cir- 
cumstances, becomes  a  huge  hygrometer. 


HYGROMETRIC    STATE   OF   THE   AIR. 


I67 


I.    Regnault's  hygrometer. 

In  this  instrument,  which  is  shewn  in  section  in  Fig.  13  (a), 
there  are  two  small  tubes  A  and  B,  the  upper  portions  being 
of  glass,  the  lower,  caps  of  silver  or  silvered  copper :  these  are 
attached  to  a  stand  so  as  to  be  near  one  another.  In  A  is 
placed  a  cork  with  a  thermometer  T' ;  in  B  a  cork  with  a  ther- 
mometer T  and  two  glass 
tubes  C,  D.  When  in  use  B 
is  partially  filled  with  alcohol 
or  ether,  and  the  tube  C  con- 
nected with  an  aspirator ;  or 
else  air  is  forced  through  the 
tube  D.  The  evaporation  of 
the  alcohol  or  of  the  ether  thus 
produced  lowers  the  tempera- 
ture of  the  metal  cap,  and  dew 
is  seen  to  form  on  the  out- 
side. 

point.     At  the  same  instant   T'  is  read,  and  the  two  tempera- 
tures T,  T'  determine  the  humidity. 


Fig.  13  («). 

The  temperature  then  shewn  by   T  is  called  the  Dew- 


Let  /  =  weight  of  moisture  in  a  given  volume  of  the  surround- 

ing air. 
P  =  weight  of  moisture  in  the  same  volume  of  the  surround- 

ing air,  supposed  to  be  saturated. 
f=  pressure  of  moisture  in/. 
F=  pressure  of  moisture  in  P. 

Then  by  what  has  already  been  shewn, 


P      F 

Now  from  Regnault's  tables  of  vapour  tension,  /  and  F  are 
known  when  we  know  T  and  T'.  Hence  the  relative  humidity  is 
at  once  given  by  these  two  temperatures.  Usually,  for  practical 


1 68 


EXPERIMENTAL    PHYSICS. 


purposes,  this  ratio  -  is  expressed  in  terms  of  a  unit  of  satu- 
ration taken  to  be  i  or  100.  Thus,  if  T=6°  C.  and  T' =  15°  C., 
the  humidity  would  be  .55,  or  55. 

Precautions.  —  In  finding  the  dew-point,  the  aspirator  is 
adjusted  so  that  the  dew  appears  and  disappears  within  small 
limits  of  temperature,  and  a  mean  taken.  The  two  caps  should 
present  the  same  appearance  and  be  placed  in  a  suitable  light. 
The  thermometers  may  be  read  by  a  telescope  placed  at  a 
distance.  In  warm  weather  alcohol  may  be  used  in  B ;  but  in 
winter,  when  the  dew-point  sometimes 
falls  below  freezing  point,  ether  must  be 
used. 

2.  AlluarcTs  hygrometer. 
This  is  an  improved  form  of  Regnault's, 
in  which  the  two  metal  caps  are  replaced 
by  a  brass  box  A  plated  with  gold,  and 
a  piece  of  the  same  material  B  (Fig.  14) 
placed  near  it,  but  not  in  contact,  so  that 
a  comparison  can  be  readily  made  between 
the  two  surfaces,  and  the  presence  of  dew 
on  A  at  once  detected.  Inserted  in  the 
box  A  is  a  thermometer  t,  which  can  be 
read  if  necessary  through  the  glass  win- 
dow in  A.  Another  thermometer  t'  gives 
the  temperature  of  the  surrounding  air. 
The  liquid  to  be  evaporated  is  poured 
in  at  E,  which  is  provided  with  a  cork 
to  prevent  evaporation,  and  two  metal 
tubes  D,  C  lead  into  the  box,  the  former  just  entering  it,  and 
the  latter  running  to  the  bottom.  Two  stopcocks  are  provided 
at  F,  G. 

To  use  the  instrument,  ether  is  poured  in  at  E  and  the  cork 
replaced  ;  the  stopcocks  F,  G  are  opened,  and  a  tube  leading 
from  an  aspirator  attached  to  /.  The  dew  is  made  to  appear 


Fig.  14. 


HYGROMETRIC   STATE  OF   THE  AIR.  169 

and  disappear  a  number  of  times,  and  a  mean  temperature 
taken,  /'  being  read  at  the  same  time.  When  finished,  one 
may  remove  the  ether  by  blowing  into  /,  and  having  another 
tube  leading  from  H  into  a  small  bottle. 

The  instrument  is  compact  and  can  easily  be  carried  about  in 
a  box ;  and  instead  of  using  an  aspirator  one  may  blow  through 
the  tube  attached  to  H  in  order  to  induce  evaporation  of  the 
ether. 

(c)    Regnaulfs  experiment. 

This  is  to  shew  that  in  a  given  space  occupied  by  a  gas  a 
liquid  will  produce  a  quantity  of  vapour  whose  weight  and  ten- 


Fig.  15. 

sion  are  the  same  as  if  the  space  were  a  vacuum  ;  or,  in  other 
words,  vapour  from  any  liquid  forms  in  presence  of  a  gas  just  as 
in  a  vacuum,  and  its  pressure  is  added  to  that  of  the  gas.  That 
the  latter  is  true  may  be  readily  shewn  by  the  apparatus  used 
for  proving  Boyle's  law  (Exp.  12,  Elementary  Course);  and  the 
following  experiment  enables  us  to  shew  that  the  weight  of 
vapour  formed  in  presence  of  a  gas  is  equal  to  that  produced  in 
vacuo. 

Fig.  15  shews  the  apparatus.     A  glass  bulb  A,  open  at  one 
end,  is  fil]ed  with  pieces  of  sponge  saturated  with  water ;  the 


1 70  EXPERIMENTAL   PHYSICS. 

other  end  fits  into  a  closed  brass  vessel  BC,  with  a  sieve  inside 
covered  with  wet  pieces  of  cotton  waste ;  another  tube  leads 
from  the  inside  of  the  sieve,  through  two  drying  tubes  D,  E,  to 
an  aspirator,  such  as  is  used  in  the  previous  experiments.  When 
the  aspirator  is  started,  air  is  drawn  through  A,  filters  through 
BC,  and  thereby  becomes  thoroughly  saturated  with  water 
vapour ;  it  then  deposits  its  moisture  in  the  drying  tubes  D,  E, 
and  enters  the  aspirator,  where  it  is  again  saturated.  A  ther- 
mometer gives  the  temperature  of  the  saturated  air. 

If  now  V=  volume  of  aspirator  in  cubic  centimeters, 

F=  tension  of  vapour,  found  from  t  and  the  tables, 
/  =  temperature  of  vapour, 

then,  if  the  space  V  were  a  vacuum,  the  weight  of  vapour  pro- 
duced in  it  in  presence  of  water  would  be 

^  x. 001 293  X. 622 
760(1  +  at} 

since  a  liter  of  dry  air  at  o°  C.  and  760  weighs  1.293  grams, 
and  the  density  of  water  vapour  is  .622. 

This  expression  should  give  the  same  result  as  would  be 
obtained  by  getting  the  difference  in  weight  of  the  drying 
tubes  before  and  after  the  experiment. 

The  weight  of  vapour  is  then  calculated  from  the  foregoing 
expression,  and  also  observed  directly  by  weighing  the  drying 
tubes  :  the  two  results  should  be  the  same. 


XV.    DENSITY    OF   A   VAPOUR    (DUMAS'   METHOD). 

This  method  is  one  of  direct  weighing,  a  glass  envelope 
being  first  weighed  when  empty,  and  then  when  filled  with  the 
vapour  whose  density  is  required ;  and  finally  its  internal  capacity 
found  by  filling  it  with  water  and  again  weighing. 

The  experiment,  though  theoretically  simple,  is  attended  with 


DENSITY   OF  A   VAPOUR. 


I/I 


Fig.   16. 


many  practical  difficulties,  and  if  performed  with  due  care  is  a 
most  instructive  exercise  for  the  student. 

The  apparatus  is  simple :  a  glass  bulb  with  a  bent  stem,  after 
being  carefully  dried  and  weighed,  has  a  small  quantity  of  the 
liquid  whose  vapour  density 
is  required  introduced  into 
it,  and  it  is  then  placed  in  a 
vessel  of  water,  as  shewn  in 
Fig.  16. 

On  heating  this  to  the  tem- 
perature at  which  the  liquid 
boils,  the  air  in  the  bulb  is 
driven  out  gradually  along 
with  the  vapour ;  and  finally, 
when  all  the  liquid  has  dis- 
appeared, the  end  of  the  stem 
is  sealed  up,  and  the  tempera- 
ture of  the  bath  and  barometric  reading  noted. 

It  is  then  removed,  dried,  and  again  weighed,  and  then,  the 
point  being  broken  off  under  water,  it  is  filled  by  water  rushing 
into  the  vacuum.  It  is  again  weighed  in  order  to  determine  its 
volume. 

For  a  very  exact  determination,  it  will  be  necessary  to  read 
the  barometer  and  hygrometer  when  weighing,  and  also  to  take 
the  temperature  in  the  balance  case. 

Theory. — 

Let /  =  weight  of  bulb  (open)  in  grams. 

p'  =  weight  of  bulb  filled  with  vapour  and  sealed. 
H=  barometric  reading  when  sealed. 
//'  =  barometric  reading  when  weighing   bulb   filled   with 

vapour. 

tj  f=  corresponding  temperature  of  room,  and  pressure  of 
vapour. 


172  EXPERIMENTAL   PHYSICS. 

VQ=  capacity  of  bulb  at  o°  C.  in  cubic  centimeters. 
T==  temperature  of  bath  at  which  bulb  is  sealed. 
k=  coefficient  of  expansion  of  glass. 
«  =  coefficient  of  expansion  of  air. 
P=.  weight  of  bulb  when  filled  with  water. 
ff"t  /',/'  =  corresponding  barometric  reading,  temperature 

of  room,  and  pressure  of  vapour. 
x=  density  of  vapour  (compared  with  dry  air  at  o°  C. 

and  760  m.m. 
d=  weight   in  grams  of   I  c.c.  of  water  at  temper- 

ature /'. 

In  the  above,  usually  H=H'  =  H",  and  /=/',  and/=/'. 
Then/'—  /  =  weight  of  vapour  less  weight  of  displaced  air  of 
room, 


and        P—p  =  weight  of  water  less  weight  of  displaced  air  of 
room, 


and  on  dividing  one  relation  by  another,  we  get  x  by  eliminat- 

ing ^o- 

These  are  rigorous  relations  from  which  x  may  be  found.  For 
ordinary  purposes  the  following  approximate  formula  may  be 
used. 

Let      /  =  weight  open. 

p'  =  weight  filled  with  vapour  and  sealed. 
P  =  weight  filled  with  water. 

/,  H—  temperature  of  bath,  and  barometric  reading  when 
sealed. 


COEFFICIENT   OF   EXPANSION   OF   METALS.  173 

\  H'= temperature   in    balance    case    and    barometric 

reading  when  weighing. 
d=  density  of  air  referred  to  water  for  f,  H'. 
x= density  of  vapour  referred  to  air  at  o°  and  760  m.m. 

ft-p     i         }H'(l+at} 

-^ •  — .-f- 1  ?  -^-=— — > 


«  =  .  003665. 


Precautions.  —  i .  The  bulb  must  be  dry,  clean,  and  carefully 
weighed  to  the  tenth  of  a  milligram. 

2.  Sufficient  of  the  liquid  whose  vapour  density  is  required 
must  be  put  in  the  bulb,  so  that  the  air  may  be  all  driven  out. 

3.  When  the  liquid  has  all  disappeared,   the   end   must  be 
sealed  and  the  temperature  of  the  bath  and  barometric  reading 
noted.     Some  difficulty  will  be  found  in  doing  this  ;  with  vapours 
such  as  chloroform  or  ether  the  escaping  vapour  may  be  burned, 
and  in  that  way  watched.     On  no  account  must  the  bulb  be 
raised  out  of  the  water  when  nearing  the  end  of  the  experiment, 
as  the  cooling  causes  air  to  be  drawn  in,  and  this  is  rarely  all 
driven  out  again. 

4.  Distilled  water  must  be  used  for  finding  the  volume.     If 
the  bulb  has  been  properly  sealed,  then,  when  the  point  is 
broken  off  under  water,  it  should  be  almost  completely  filled, 
leaving  only  a  small  air  bubble  about  a  centimeter  in  diameter. 
For  ordinary  volatile  vapours  a  water  bath  is  used ;   for  iodine 
vapour,  sulphuric  acid  or  olive  oil ;  and  if  a  higher  temperature 
is  required,  some  kind  of  metallic  alloy  may  be  used. 

XVI.     COEFFICIENT   OF   EXPANSION   OF   METALS. 

The  most  accurate  method  for  determining  a  coefficient  of 
expansion  of  a  metal  is  that  devised  by  Ramsden,  in  which  two 
microscopes,  fixed  in  position,  are  provided  with  micrometer 
eyepieces  and  movable  cross-wires,  the  micrometer  head  being 
graduated  so  that  each  division  corresponds  to  a  very  small 


1/4  EXPERIMENTAL   PHYSICS. 

fraction  of  an  inch.  The  metal  to  be  examined,  made  into  a 
suitable  bar,  is  placed  under  the  microscopes,  which  are  so 
adjusted  laterally  that  the  cross-wires  coincide  with  certain 
fixed  marks  on  the  bar  at  a  certain  temperature. 

As  this  temperature  is  increased  or  diminished,  the  fixed 
marks  are  seen  to  change  position  when  viewed  through  the 
microscopes  ;  and  the  corresponding  increase  or  diminution  in 
length  can  easily  be  calculated  from  the  number  of  turns  of  the 
micrometers  necessary  to  bring  the  cross-wires  again  in  coin- 
cidence with  the  fixed  marks.  The  method  is  simple,  satisfac- 
tory, and,  with  due  precaution,  very  exact. 

Usually,  for  the  purposes  of  comparing  standards  of  length 
and  finding  their  coefficients,  a  machine  known  as  a  compara- 
tor is  used,  which  consists  essentially  of  two  microscopes  as 
described  above,  and  special  arrangements  with  rollers  to  level 
the  bars  or  to  raise  or  lower  them :  some  way  must  also  be 
found  for  maintaining  them  at  constant  temperatures. 

The  bars  may  be  made  of  any  form,  and  fine  lines  ruled  upon 
them  with  a  dividing  engine ;  but  for  standards  of  length  they 
are  made  of  square  cross-section,  and  near  each  end  are  drilled 
two  circular  openings  to  the  mid-depth  of  the  bar.  On  the 
bottoms  of  these  surfaces  lines  are  engraved,  which  thus  lie  in 
the  neutral  axis  of  the  bar,  and  are  not  so  liable  to  change  their 
position  owing  to  flexure  or  strain. 

XVII.    SPECIFIC  HEAT  OF  DRY  AIR  AT  CONSTANT  PRESSURE. 

Regnault's  experimental  determination  of  this  is  the  most 
reliable.  His  method,  which  is  one  of  mixture,  consists  in 
heating  a  given  mass  of  air  to  a  high  temperature,  and  then 
allowing  it  to  pass  at  a  constant  pressure  (nearly  atmospheric) 
into  a  calorimeter  containing  a  known  quantity  of  water.  The 
equivalence  of  heat  lost  and  gained  determines  the  specific  heat 
of  the  air. 

An  apparatus  similar  to  that  devised  by  Regnault  is  shewn  in 
Fig.  17. 


SPECIFIC    HEAT   OF    DRY    AIR. 


175 


A  large  vessel  R  filled  with  water  contains  an  inner  reservoir 
of  about  35  liters'  capacity,  in  which  dry  air  is  compressed  by 
means  of  an  air  pump  attached  at  A.  A  manometer  at  M, 
which  may  be  of  any  kind  (although  for  exact  results  a  mercurial 
one  is  preferable),  gives  the  pressure.  Leading  from  this  reser- 
voir is  a  tube  which  passes  to  a  heating  apparatus  C,  runs 
through  it  in  a  spiral  form,  and  emerges  at  the  bottom,  and 
finally  leads  into  a  small  calorimeter  c.  A  conical  stopcock  V 
regulates  the  flow  of  air  into  C,  and  by  watching  a  small  water 


Fig.  17. 

gauge  G  the  operator  can  maintain  a  pressure  which  is  never 
very  much  greater  than  atmospheric,  and  is  perfectly  steady. 
A  stirrer  5  and  thermometer  are  placed  in  C,  and  a  stirrer  s 
and  thermometer  /  in  c.  The  air  emerges  eventually  at  e. 

Theory.  — 

.    Let  H=  total  pressure  of  air  in  reservoir. 
V—  capacity  of  reservoir  in  liters. 
//  =  pressure  of  barometer. 
T=  temperature  in  C. 


1/6  EXPERIMENTAL   PHYSICS. 

/  =  initial  temperature  in  c. 
f'  =  final  temperature  in  c. 
W  =  weight  of  water  in  c. 
w  =  water  equivalent  of  the  calorimeter. 
x—  specific  heat  of  air  at  the  mean  pressure  observed. 

Then,  after  the  experiment  is  finished,  there  will  be  a  quan- 
tity of  air  left  in  the  reservoir  equal  to  V,  the  pressure  of 
which  is  h ;  and  therefore  by  Boyle's  law  the  volume  which 
has  escaped  is 


h 

And,  since  one  liter  weighs  1.293  grams,  weight  of  air 
=  — j— -  V  x  1.293  grams, 

and,  by  the  equivalence  of  heat  lost  and  gained, 
H  —  h      jT-  .  . 


Precautions.  — There  are  many  corrections  to  be  made  in  this 
formula  if  an  accurate  result  is  required.  The  air  in  the  reser- 
voir should  be  dry ;  if  not,  then  a  complicated  correction  is  nec- 
essary, both  for  temperature  and  pressure  of  vapour  contained 
in  it.  Evidently  h  will  be  corrected  by  the  mean  reading  of  G; 
and  the  usual  precautions  with  regard  to  maintaining  a  constant 
temperature  in  C,  c,  by  use  of  the  stirrers,  and  also  to  insure 
equal  radiation  and  absorption  in  c,  must  be  taken. 

XVIII.     PRESSURE  OF  VAPOURS  FOR  LOW  TEMPERATURES. 

(i)     Between  zero  and  100°  C. 

The  apparatus  for  this  is  that  devised  by  Dalton  and  modified 
by  Regnault  :  it  is  shewn  in  the  following  figures. 

A  barometer  A  is  placed  alongside  of  another  tube  B  which 


PRESSURE   OF  VAPOURS   FOR   LOW   TEMPERATURES.      177 


Fig.  19. 


Fig.   18. 


has  an  attachment  at  its  upper  end  shewn  in  Fig.  19 :  this  is 
simply  a  small  glass  globe  which  is  detachable  at  D  and  provided 
with  a  stopcock  7!  A  vessel  C  con- 
taining water,  a  stirrer,  and  a  ther- 
mometer, surrounds  the  upper  part  of 
A  and  B. 

When    in    use,    the    liquid   whose 
vapour  tension  is  to  be  examined  is 

placed  in  G,  and  an  air 

pump  being  attached, 

exhaustion    is    carried 

on  until  only  vapour  is 

left,    and     T   is    then 

closed.        The     differ- 
ences in  the  readings 

of  A  and  B  at  different 
temperatures  give  the  required  pressures. 
(2)     For  the  temperatures  below  zero. 

For  temperatures  below  the  freezing  point  of  water  the  appara- 
tus is  slightly  altered  as  in  Fig.  20. 
A  and  B  are  two  barometer 
tubes  :  at  the  upper  end  of  B  is  a 
glass  envelope  G  in  which  is  the 
liquid,  so  that  the  space  above 
the  mercury  in  B  is  filled  with 
vapour.  Surrounding  G  is  a  freez- 
ing mixture  (liquid)  contained  in  a 
vessel  C.  The  tension  of  the 
vapour  can  thus  be  found  for  tem- 
peratures below  freezing  point, 
and,  since  the  two  mercurial  por- 
tions are  exactly  in  the  same 
conditions,  the  tension  is  given 
directly  by  the  difference  in  level 
between  A  and  B. 


Fig.  20. 


178  EXPERIMENTAL    PHYSICS. 

The  method  is  exact,  if  one  takes  care  to  insure  the  absence 
of  air  in  the  tube  B,  which  might  be  done  more  readily  by  using 
an  arrangement  provided  with  a  stopcock,  as  in  Fig  19. 


XIX.     REGNAULT'S    APPARATUS    FOR   THE    DETERMINATION 
OF  THE  PRESSURE  OF  VAPOURS  AT  HIGH  TEMPERATURES. 

This  is  represented  in  Fig.  21.  A  closed  boiler  B  contains 
water  (or  other  liquid),  the  temperature  of  which  is  given  by  the 
thermometer  t.  A  large  reservoir  R  is  connected  to  the  boiler, 
and  a  compression  pump  at  A,  with  a  manometer  at  M,  com- 
pletes the  arrangement.  A  water  jacket  CD  surrounds  the 


Fig.  21. 

tube  leading  from  B  to  R,  so  that  the  steam  condenses  and  runs 
back  into  the  boiler. 

The  mode  of  operation  is  to  compress  the  air  in  R,  deter- 
mine the  pressure  by  means  of  the  manometer  and  barometer, 
and  finally  take  the  temperature  indicated  by  /.  Tables  are  then 
constructed,  which  may  be  graphically  arranged  to  exhibit  the 
relation  between  temperature  and  pressure. 


ELECTRICITY   AND    MAGNETISM. 


LIST   OF   EXPERIMENTS. 

PACK 

I.   MAGNETIC  LINES  OF  FORCE 181 

II.   DETERMINATION  OF  MAGNETIC  MOMENTS         .        .        .  184 

III.  THE  DECLINATION  COMPASS 187 

IV.  THE  INCLINATION  COMPASS 190 

V.   DETERMINATION   OF   THE   INTENSITY   OF   THE    EARTH'S 

MAGNETIC  FIELD          .        .        .        .        .        .        .  195 

VI .   MAGNETIC  FIELD  OF  A  CURRENT  —  BIOT  AND  SAVART'S 

LAW 205 

VII.   MEASUREMENT  OF  CURRENT  INTENSITY   ....  210 

(1)  The  Tangent  Galvanometer 

(2)  The  Sine  Galvanometer         ..... 

VIII.   HYDROGEN  VOLTAMETER 216 

IX.   COPPER  VOLTAMETER 219 

X.   CALIBRATION  OF  GALVANOMETERS 222 

XL   GALVANIC  BATTERIES 227 

XII.   DETERMINATION  OF  RESISTANCE 239 

XIII.  TEMPERATURE  COEFFICIENT  OF  RESISTANCE   .        .        .  249 

XIV.  GALVANOMETER  RESISTANCE     ......  255 

XV.   RESISTANCE  OF  BATTERIES  OR  CELLS      ....  256 

XVI.   DETERMINATION  OF  ELECTROMOTIVE  FORCES  .        .        .  261 

XVII.    ABSOLUTE  DETERMINATION  OF  RESISTANCE  BY  USE  OF 

CALORIMETER              _.     ' 271 

179 


ELECTRICITY   AND    MAGNETISM. 

I.     MAGNETIC   LINES   OF   FORCE. 

When  a  magnetized  steel  bar  is  plunged  into  a  vessel  con- 
taining iron  filings,  it  is  found  on  withdrawing  it  that  these 
arrange  themselves  in  tufts  or  bunches,  not  along  the  whole 
length  of  the  bar,  but  round  its  ends,  leaving  the  central  por- 
tion quite  free. 

From  this  it  would  appear  that  the  centers  of  attraction,  or, 
as  they  are  commonly  called,  the  magnetic  masses,  are  situated 
in  the  regions  near  the  ends  of  a  magnet,  and  it  is  customary 
to  call  the  points  in  these  regions  to  which  the  attractions  are 
the  greatest  the  poles  of  the  magnet,  and  the  straight  line 
joining  them  its  axis. 

If  a  bar  magnet  be  suspended  so  that  it  can  move  freely  in  a 
horizontal  plane,  it  will  take  up  a  position  in  space  pointing 
nearly  north  and  south,  and  if  it  is  displaced  from  this  posi- 
tion, it  will  come  back  to  it  and  will  always  present  the  same 
end  towards  the  north.  For  this  reason  the  end  of  a  magnet 
which  points  to  the  north  is  called  the  north  or  north-seeking 
pole,  and  the  opposite  end  the  south  or  south-seeking  pole. 
The  terms  positive  and  negative  are  also  frequently  used  to 
designate  the  same  poles. 

Further,  if  the  north  pole  of  a  second  magnet  is  presented 
to  each  of  the  poles  of  the  suspended  one,  the  north  pole  of 
the  latter  will  be  repelled  and  its  south  attracted.  Hence, 
the  law  of  magnetic  action  :  Two  magnetic  poles  of  the  same 
kind  repel,  and  two  poles  of  the  opposite  kind  attract  each 
other. 

181 


I  82 


EXPERIMENTAL   PHYSICS. 


If  a  bar  magnet  be  placed  on  a  piece  of  cork  or  wood,  and 
the  whole  be  then  floated  on  the  surface  of  water  or  mercury, 
it  will  be  found  that  under  the  action  of  the  earth  the  magnet 
is  subjected  to  a  pure  rotation,  and  not  to  any  motion  of  trans- 
lation. As  the  action  of  the  earth  may  be  taken  to  be  con- 
stant in  magnitude  and  direction  for  points  near  the  floating 
magnet,  and  since  all  the  forces  acting  on  the  latter  are  thus 
reduced  to  a  couple,  we  conclude  that  positive  and  negative 
magnetic  masses  are  present  in  equal  quantities  in  every 
magnet. 

Again,  if  we  attempt  to  separate  the  two  poles  of  a  magnet 
by  breaking  it  into  two  pieces,  we  find  that  instead  of  accom- 
plishing this,  we  only  produce  two  magnets  in  place  of  one, 
and  no  matter  into  how  many  pieces  a  magnet  is  divided,  each 

of  the  fragments  will  pos- 
sess polarity  just  as  the  mag- 
net did  as  a  whole.  We 
therefore  conclude  that  it  is 
impossible  to  obtain  a  quan- 
tity of  positive  magnetism 
which  is  not  associated  with 
an  equal  quantity  of  the 
opposite  kind.  Although 
we  cannot  obtain  a  magnet 
with  only  one  pole,  it  is 
often  useful  in  facilitating 
calculation  to  consider  a 
single  pole  as  existing  by 
itself,  always  remembering, 
of  course,  that  it  is  a  purely 
mental  conception. 

The  magnetic  mass  of  a 
unit  pole  is  defined  to  be  such  that  if  concentrated  in  a  point 
and  placed  at  a  unit  distance  from  that  of  a  similar  pole,  it 
will  repel  the  latter  with  unit  force. 


MAGNETIC   LINES   OF   FORCE.  ^3 

The  field  of  a  magnet  is  the  region  in  which  it  exerts  any 
force,  and  the  intensity  of  a  magnetic  field  at  any  point  is  the 
force  exerted  by  the  magnet  on  a  unit  pole,  supposed  placed 
at  this  point.  If  we  were  to  imagine  a  free  unit  pole  placed 
in  a  magnetic  field,  and  moving  under  the  forces  exerted  by 
the  magnet,  the  path  which  it  would  describe  is  called  a  line 
of  force,  and  it  is  such  that  the  tangent  to  it  at  each  point 
is  the  direction  of  the  resultant  force  at  that  point.  The  lines 
of  force  for  a  given  magnet  may  be  plotted  by  placing  a  very 
short  compass  needle  at  various  points  in  its  field,  and  noting 
in  what  direction  the  needle  points  while  in  each  position. 

The  actions  of  the  magnet  on  the  positive  and  negative  poles, 
respectively,  of  such  a  needle  will  be  equal  in  magnitude  and 
opposite  in  direction,  and  therefore  the  direction  which  the 
needle  takes  up  is  that  of  the  resultant  force  at  the  point 
where  it  is  placed. 

Probably  one  of  the  best  ways  to  observe  these  lines  of  force 
is  to  place  a  plate  of  glass  over  a  magnet  in  contact  with  it, 
and  then  to  sprinkle  a  quantity  of  very  fine  iron  filings  over 
the  plate.  On  gently  tapping  the  glass,  the  small  particles, 
which  act  like  so  many  compass  needles  under  these  circum- 
stances, will  arrange  themselves  under  the  influence  of  the 
magnetic  forces  into  curves  which  coincide  with  the  lines  of 
force.  If  it  is  desired  to  preserve  copies  of  such  curves,  the 
glass  plate  may  be  replaced  by  a  sheet  of  white  paper  coated 
over  with  a  very  thin  layer  of  paraffine.  If  this  paper  be  then 
heated  without  disturbing  the  filings,  these  will  sink  into  the 
melted  paraffine,  and  on  its  cooling  will  remain  embedded  in 
it.  Copies  of  this  may  then  be  taken  by  the  ordinary  photo- 
graphic processes. 

A  better  method  is  to  use  blue  printing  paper  in  place  of 
that  coated  with  paraffine,  care  being  taken  while  exposing  it 
to  the  sunlight  not  to  disturb  the  filings.  A  still  better  method 
is  to  use  an  ordinary  sensitized  photographic  plate,  and,  as 
with  the  blue  paper,  to  perform  the  experiment  in  a  darkened 


!84  EXPERIMENTAL   PHYSICS. 

room.  The  curves  formed  by  the  particles  may  be  impressed 
on  the  plate  by  illuminating  it  with  an  electric  spark,  or  by 
quickly  turning  on  and  off  the  light  from  an  incandescent 
lamp.  From  a  plate  prepared  in  this  way  any  number  of 
prints  can  be  readily  taken. 

A  diagram  of  the  lines  of  force  taken  by  this  method  is 
exhibited  in  Fig.  i,  in  which  case  they  are  due  to  the  com- 
bination of  two  dissimilar  poles.  The  student  will  find  it 
exceedingly  interesting  and  instructive  to  make  for  himself  a 
set  of  plates  showing  the  lines  of  force  produced  by  (i)  an 
ordinary  bar  magnet,  (2)  a  horseshoe  magnet,  (3)  various 
irregular  combinations  of  magnetic  poles,  (4)  the  pieces  of  a 
broken  bar  magnet  placed  close  to  each  other,  and  (5)  the  pres- 
ence of  a  piece  of  soft  iron  near  the  poles  of  a  magnet. 

From  an  inspection  of  Fig.  i  it  can  readily  be  seen  that  the 
lines  of  force  are  closer  together  near  the  poles  of  the  magnet 
than  they  are  at  a  distance  from  it.  As  the  force  exerted  by 
the  magnet  on  a  unit  pole  decreases  with  the  distance  of  the 
pole  from  the  magnet,  we  may  consider  the  number  of  lines  of 
force  perpendicularly  crossing  a  given  small  area,  taken  at  any 
point  in  the  field,  to  be  proportional  to  the  force  intensity  at 
that  point.  In  a  field  of  uniform  intensity,  therefore,  the  lines 
of  force  will  be  so  distributed  that  the  same  number  will  cut 
any  such  area  placed  as  indicated. 

II.     DETERMINATION   OF   MAGNETIC   MOMENTS. 

From  an  inspection  of  its  lines  of  force  it  is  evident  that  a 
bar  magnet  cannot  be  considered  as  having  a  single  pair  of 
poles  or  centers  of  attraction.  In  fact,  such  a  magnet  possesses 
an  indefinite  number  of  pairs  of  poles,  those  of  each  pair  being 
of  equal  strength  and  of  opposite  sign. 

If,  then,  a  bar  magnet  be  suspended  in  a  uniform  horizontal 
magnetic  field,  similar  to  that  represented  in  Fig.  2,  where 
P  denotes  the  direction  towards  which  a  north  pole  tends  to 


DETERMINATION    OF   MAGNETIC   MOMENTS. 


I85 


move,   it  will   be   acted  on  by  a   number   of   pairs  of   forces, 
whose  resultant  effect  is  that  of  a  couple,  and  under  the  action 


Fig.  2. 

of  this  couple  the  magnet  will  tend  to  take  up  a  position  such 
that  its  axis  coincides  with  the  direction  of  the  lines  of  force 
of  the  field. 

If  we  represent  the  mass- 
es of  the  various  pairs  of 
poles  by  ^,  -^;  /42,  -yu2 ; 
etc.,  and  the  distances  be- 
tween those  forming  each 
pair  by  lv  /2,  etc.,  and  if  H 
is  the  intensity  of  the  mag- 
netic field,  the  turning  cou- 
ple or  moment  of  the  forces 
about  the  point  of  suspension 
is  equal  to  /f  •  2/*/ •  sin  0, 
the  angle  Q  being  that  which 
the  magnet  makes  in  any 
position  with  the  direction 
of  the  lines  of  force. 

The  expression  2/^/,  which 
is  generally  written  M,  is 
a  constant  for  each  magnet, 
irrespective  of  how  it  may 
be  situated,  and  is  called  its 

Fig.  3. 

magnetic  moment. 

One  of  the  best  methods  for  comparing  the  magnetic  moments 


lS6  EXPERIMENTAL   PHYSICS. 

of  different  magnets  is  that  of  the  torsion  balance,  and  in  case 
absolute  determinations  are  required  these  can  be  made  with 
a  fair  degree  of  accuracy  by  means  of  the  same  instrument. 

Such  a  balance  as  ordinarily  used  (Fig.  3)  consists  of  a 
glass  case  provided  with  a  scale  placed  round  its  sides,  and 
graduated  so  as  to  indicate  degrees.  Situated  in  the  same 
plane  as  the  scale  is  the  magnet  to  be  tested.  It  is  suspended 
by  a  fine  wire  which,  after  passing  through  a  glass  tube,  C, 
attached  perpendicularly  to  the  cover  of  the  case,  has  its 
upper  end  fastened  to  a  brass  torsion  head,  A.  This  piece 
rests  on  a  graduated  brass  collar,  B,  surrounding  the  upper 
portion  of  C,  and  by  means  of  these  graduations,  and  a  mark 
on  the  torsion  head,  any  angle  through  which  the  latter  may 
be  turned  can  be  directly  ascertained. 

Having  placed  the  balance  in  some  suitable  position,  it  is 
first  necessary  to  find  what  two  graduations  on  the  scale  lie  in 
the  earth's  magnetic  meridian.  This  is  best  done  by  placing  a 
compass  needle,  resting  easily  on  a  pivot,  at  the  center  of  the 
case,  or  by  suspending  it  from  A  by  a  torsionless  fiber,  such  as 
a  single  strand  of  silk.  The  needle  will,  under  these  conditions, 
lie  in  the  meridian,  and  the  graduations  on  the  scale  towards 
which  it  points  may  then  be  easily  noted. 

A  bar  of  brass  or  copper  of  about  the  same  dimensions  as 
the  magnet  to  be  examined,  should  then  be  suspended  by  the 
wire,  and  the  torsion  head  rotated  until  this  bar  rests  in  the 
meridian.  If  the  bar  of  brass  or  copper  be  then  detached 
without  disturbing  the  wire,  and  the  magnet  put  in  its  place, 
the  latter  will,  after  oscillating  for  some  time,  finally  come  to 
rest  in  the  meridian  without  being  subjected  to  any  torsional 
strain.  If  then  A  be  turned  through  some  angle  «,  the  magnet 
will  be  displaced  from  the  meridian  by  an  angle  9,  and  (a  —  0), 
which  is  generally  denoted  by  <£,  will  be  the  amount  of  torsion 
in  the  wire. 

When  the  magnet  is  in  this  position  the  torsion  couple  is 
balanced  by  the  action  of  the  forces  due  to  the  earth's  magnetic 


THE   DECLINATION   COMPASS. 


I87 


field,  and  since  this  may  be  taken  to  be  uniform,  we  have,  by 
taking  moments  about  the  point  of  suspension,  the  relation 

HMs'm0=C<f>, 

"•       _         "=ra: 

H  being  the  horizontal  intensity  of  the  earth's  field,  M  the  mag- 
netic moment,  and  C  the  constant  of  torsion  for  the  wire.  (See 
Appendix  B  for  method  of  determining  C.) 

A  method  of  finding  H  is  given  in  Experiment  V.,  and  as 
both  this  quantity  and  C  can  be  ascertained  by  preliminary 
investigations,  it  only  remains  to  determine  an  accurate  value 

for-*-- 
sin  0 

This  is  done  simply  by  repeating  the  experiment  just  de- 
scribed, a  number  of  times,  care  being  taken  at  each  trial  to 
give  to  A  a  different  amount  of  rotation. 

If  it  is  desired  to  make  a  comparison  of  two  magnets,  it  is 

only  necessary  to  determine  a  value  for  for  each  of  them, 

sintf 

provided  the  same  torsion  wire  is  used  and  both  experiments 
are  conducted  in  the  same  place.  The  ratio  of  these  two  quan- 
tities will  be  equal  to  that  of  the  two  magnetic  moments,  and  if 
one  of  them  is  known,  the  other  can  then  be  found  in  terms  of  it. 

The  method  is  especially  suitable  for  magnets  which  are  long 
and  slender.  In  case  they  are  very  short  the  angular  deflec- 
tions can  be  measured  with  much  greater  accuracy  by  adopting 
the  method  exhibited  in  Fig.  13. 

In  this  experiment  the  best  results  are  obtained  by  using  fine 
silver  wire  or  quartz  fibers  for  the  suspensions. 

III.     THE    DECLINATION    COMPASS. 

It  has  already  been  pointed  out  that  if  a  bar  magnet  be  sus- 
pended so  as  to  rotate  freely  in  a  horizontal  plane,  it  will  take 
up  a  position  with  its  axis  pointing  in  a  northerly  direction. 


1 88 


EXPERIMENTAL    PHYSICS. 


In  Experiment  I.,  it  has  also  been  shewn  that  the  field  of  a 
magnet  can  be  explored  by  a  small  compass  needle  being  placed 
at  different  points  in  it,  since  the  needle  will  come  to  rest  in 
each  position  with  its  axis  always  coinciding  with  the  direction 
of  the  lines  of  force. 

/Such  considerations  as  these  lead  us  to  the  conclusion  that 
the  earth  is  an  immense  magnet  surrounded  with  a  field  of 

force,  and  that  therefore 
the  bar  magnet  suspended 
as  above  comes  to  rest 
pointing  towards  the  north 
simply  because  it  performs 
the  same  function  with 
regard  to  the  earth's  field 
that  the  small  compass 
needle  does  with  regard 
to  that  of  a  given  magnet. 
In  order  to  obtain  exact 
knowledge  of  the  earth's 
magnetic  action  at  a  point, 
it  is  necessary  to  deter- 
mine (i)  the  direction  of 
the  lines  of  force  and  (2) 
the  intensity  of  the  mag- 
netic action  at  that  point. 
The  directions  of  ter- 
restrial lines  of  force  are 
defined  with  respect  to  the  geographical  meridian,  and  to  the 
horizontal  plane  through  any  point.  The  vertical  plane  passing 
through  the  axis  of  a  freely-suspended  horizontal  magnet  at  rest 
is  called  the  magnetic  meridian,  and  the  angle  between  it  and 
the  geographical  meridian,  the  declination. 

Since  the  magnetic  poles  of  the  earth  lie  in  the  magnetic 
meridian,  the  lines  of  force  at  any  place  will  be  parallel  to  this 
plane,  and,  therefore,  the  first  step  to  take  in  ascertaining  their 


Fig.  4. 


THE    DECLINATION    COMPASS.  189 

direction,  is  to  make  an  accurate  determination  of  the  declina- 
tion. 

This  is  found  by  means  of  an  instrument  called  the  declina- 
tion compass.  It  consists,  as  represented  in  Fig.  4,  of  a  circular 
brass  or  copper  box  AB,  attached  at  right  angles  to,  and  freely 
rotating  about,  a  foot  P,  which  is  itself  supported  by  a  tripod. 
On  the  bottom  of  this  box  there  is  placed  a  divided  circle  M, 
at  whose  center  a  lozenge-shaped  compass  needle  is  delicately 
pivoted.  To  the  box  there  are  fastened  two  uprights  supporting 
a  horizontal  axis  X,  to  which  is  fixed  an  astronomical  telescope, 
movable  in  a  vertical  plane.  The  foot  P  also  carries  a  fixed 
azimuthal  circle  QR,  and  by  means  of  it  and  the  vernier  Kany 
rotation  given  to  the  box  may  be  determined.  The  graduated 
arc  x,  attached  to  the  upright  and  the  vernier  K,  which  moves 
with  the  telescope,  afford  a  means  of  measuring  any  angle 
through  which  the  latter  may  be  turned. 

In  determining  the  declination-,  the  divided  scale  in  the  bottom 
of  the  box  is  first  made  horizontal  by  using  the  level  suspended 
from  the  telescope's  support  as  an  indicator,  and  the  screws  in 
the  tripod  for  making  the  adjustment.  If  the  geographical 
meridian  through  the  point  of  observation  has  been  previously 
determined,  the  telescope  is  then  sighted  upon  some  distant 
mark  that  also  lies  in  it. 

Since  the  divided  scale  is  so  placed  that  its  zero  diameter  is 
in  the  same  vertical  plane  as  the  axis  of  the  telescope,  it  will 
also  lie  in  the  geographical  meridian  when  this  is  done,  and  the 
angle  between  it  and  the  axis  of  the  compass  needle  in  its  posi- 
tion of  rest  will  be  the  declination. 

In  case  the  geographical  meridian  has  not  been  located,  it  will 
be  necessary  to  sight  the  telescope  upon  the  sun  at  noon,  or  on 
some  star  whose  time  of  transit  is  given  in  the  Nautical  Almanac. 
When  the  telescope  is  so  adjusted,  the  diameter  of  the  azimuthal 
circle  M  lying  in  the  meridian  can  then  be  ascertained,  and  the 
declination  will  as  before  be  the  angle  between  the  axis  of  the 
compass  needle  and  this  diameter.  As  it  frequently  happens 


190 


EXPERIMENTAL    PHYSICS. 


that  owing  to  imperfect  magnetization  the  magnetic  axis  does 
not  coincide  with  the  geometrical  axis  of  the  needle,  it  should 
be  inverted  so  that  what  was  its  lower  face  in  the  first  case 
will  be  the  upper  in  the  second.  The  mean  of  the  two  readings 
obtained  should  be  taken  for  the  declination,  and  the  whole 
operation  repeated  several  times  to  insure  accuracy. 

The  declination  is  said  to  be  east  or  west  according  as  the 
north  pole  of  the  needle  is  to  the  east  or  west  of  the  geograph- 
ical meridian.  The  results  obtained  from  a  series  of  observa- 
tions, extending  over  a  long  term  of  years,  indicate  a  gradual 
change  at  each  point,  the  declination  being  at  one  time  east  or 
west,  and  after  increasing  up  to  a  certain  limit,  then  diminishing 
and  passing  to  the  other  side  of  the  meridian.  Besides  this 
change,  which  is  very  slow,  there  are  yearly  and  daily  variations, 
and,  in  addition  to  these,  abrupt  changes  frequently  occur  owing 
to  the  existence  of  so-called  magnetic  storms. 

Maps  are  issued  each  year  from  various  magnetic  observa- 
tories, shewing  the  declination  at  various  places,  and  on  these 
are  drawn  lines,  called  isogonic,  which  pass  through  those  points 
where  the  declination  is  the  same. 

IV.     THE   INCLINATION    COiMPASS. 

Having  determined  the  magnetic  meridian,  or  plane,  parallel 
to  the  lines  of  force,  it  then  remains  to  find  the  angle  their 
direction  makes  with  the  horizontal.  This  angle  is  called  the 
inclination  or  dip,  and  is  determined  by  the  inclination  compass 
or  dipping  needle. 

A  common  form  of  the  instrument  is  exhibited  in  Fig.  5, 
where  »/  is  an  azimuthal  circle  supported  on  three  feet  supplied 
with  leveling  screws,  and  A  is  a  plate  which  carries  a  spirit 
level,  and  is  free  to  rotate  about  a  vertical  axis.  A  frame  r, 
resting  on  two  uprights,  supports  the  graduated  circle  M  in  a 
vertical  plane,  and  ab,  a  magnetic  compass  needle,  rotates 
about  a  horizontal  axis  through  its  center  of  gravity,  the  resting 


THE   INCLINATION   COMPASS. 


191 


points  being  pieces  of  agate  embedded  in  the  frame  at  the 
center  of  the  vertical  circle.  The  instrument  is  so  constructed 
that  when  the  plate  A  is 
level,  the  zero  diameter  of 
the  circle  M  is  horizontal. 

In  determining  the  incli- 
nation, the  instrument  must 
first  be  leveled  and  then  any 
of  the  following  methods 
may  be  applied  : 

METHOD  I.  —  Turn  the 
frame  A  very  slbwly  about 
its  vertical  axis  of  rotation, 
and  observe  the  values  of 
the  inclination  during  the 
motion.  The  greatest  dip 
noted  will  be  the  true  incli- 
nation at  the  point  of  obser- 
vation, as  it  is  obtained  when 
the  vertical  circle  is  in  the  magnetic  meridian,  and  the  needle  is 
then  acted  upon  by  the  total  resultant  of  the  earth's  magnetic 
action. 

METHOD  II. — Rotate  the  plate  A  until  the  needle  assumes 
a  vertical  position  and  points  towards  the  ninety-marks  on  the 
circle  M.  Since  the  needle  is  suspended  at  its  center  of 
gravity,  and  in  this  position  is  acted  upon  only  by  the  vertical 
component  of  the  terrestrial  field,  the  circle  Mmust  then  be  at 
right  angles  to  the  magnetic  meridian.  By  next  turning  it 
through  a  right  angle,  as  indicated  by  the  azimuthal  circle  ?#, 
the  needle  will  be  in  the  meridian,  and  the  inclination  may  be 
noted. 

METHOD  III. — By  this  method  the  vertical  circle  is  placed 
successively  in  two  planes  which  are  at  right  angles  to  each 
other,  and  are  situated  one  on  each  side  of  the  meridian.  The 
angle  between  the  needle  and  the  horizontal  diameter  is  noted 


Fig.  5. 


192 


EXPERIMENTAL   PHYSICS. 


in  each  case,  and  from  these  angles  the  inclination  at  the  point 
of  observation  is  determined. 

The  disposition  is  that  shewn  in  Fig.  6.  AHB  represents 
the  magnetic  meridian,  AEB  a  plane  making  an  angle  6  with 
it,  and  ACB  a  plane  at  right  angles  to  AEB.  D1  and  D2 


denote  the  angles  made  by  the  needle  with  the  zero  diameter 
of  the  vertical  circle  when  it  is  at  rest  in  each  of  these  positions. 

If  D  is  the  true  value  of  the  inclination  at  points  in  the 
immediate  neighbourhood  of  the  observing  station,  and  F  is  the 
resultant  intensity  of  the  earth's  magnetic  action  on  the  needle 
there,  the  latter  can  be  resolved  into  two  components  in  the 
magnetic  meridian  :  one  horizontal  FcosD,  and  the  other  verti- 
cal FsinD.  The  component  FcosD  can  again  be  resolved  in 
an  infinite  number  of  ways  into  two  others  in  the  horizontal 
plane.  In  a  direction  making  an  angle  6  with  the  meridian,  it 
has  a  component  Fcos  D  cos  6,  and  in  one  perpendicular  to  this, 
a  component  FcosD  sin  0.  These  forces  are  represented  in 
the  figure  as  acting  on  the  needle  when  it  is  at  rest  in  the 
plane  AEB. 

As  the  force  FcosD  sm  6  acts  perpendicularly  to  this  plane, 
the  position  occupied  by  the  needle  is  not  affected  by  it,  and 


THE    INCLINATION   COMPASS.  !93 

the  angle  Dl  is  therefore  completely  determined  by  the  relation 


*'-       ^sin^      ' 
or  cot  Dl  =  cot  D  cos  6.  (i) 

From  similar  considerations  it  can  be  readily  seen  that  when 
the  vertical  circle  is  turned  into  the  plane  ACB  the  component 
FcosDcosQ  does  not  affect  the  displacement  of  the  needle, 
and  the  angle  Dz  is  therefore  given  by 

cot  Z>2  =  cot  D  sin  6.  (2) 

Eliminating  6  from  (i)  and  (2),  it  follows  that 
cot2  D  =  cot2  £>!  +  cot2  Z>2. 

If,  therefore,  the  apparent  inclinations  D±  and  Z>2  are  found 
in  any  two  planes  at  right  angles  to  each  other,  the  true  inclina- 
tion can  be  calculated  by  applying  this  relation. 

METHOD  IV.  —  If  when  the  divided  circle  M  is  in  the  mag- 
netic meridian  the  needle  be  given  a  slight  displacement  from 
its  position  of  rest,  it  will  oscillate  harmonically,  and  if  F  is 
the  intensity  of  the  earth's  field,  M1  the  magnetic  moment  of 
the  needle,  and  M2  its  moment  of  inertia  about  the  axis  of  rota- 
tion, we  have  the  time  of  a  small  oscillation  given  by 

/1  =  2 
since  the  equation  of  motion  is 


If  the  circle  be  now  turned  so  that  its  plane  is  at  right 
angles  to  the  magnetic  meridian,  and  the  needle  be  again 
made  to  perform  small  oscillations,  its  periodic  time  is  given 

by  /2  =  2?r\  ?       ,  since  in  this  case  the  vertical  compo- 

nent of  the  earth's  attraction  alone  is  acting. 


i94 


EXPERIMENTAL    PHYSICS. 


t  2 
We  have,  therefore,   sin  D=^   and  from  this  equation  the 

inclination  may  be  calculated.  The  times  t±  and  f2  should  be 
found  by  allowing  the  needle  to  oscillate  for  a  considerable 
time. 

Method  III.  will  probably  give  the  most  satisfactory  results, 
but  the  sources  of  error  are  so  numerous  in  any  case  that  the 
greatest  care  must  be  taken  in  making  the  adjustments  if  accu- 
rate determinations  are  required. 

Precautions.  —  I.  It  frequently  happens  that  from  various 
causes  the  vertical  circle  becomes  displaced  so  that  the  zero 
diameter  is  not  horizontal  when  the  plate  A  is  level.  It  then 
becomes  necessary  to  ascertain  the  true  horizontal  diameter. 

Since  the  needle  will  be  vertical  when  its  plane  of  rotation  is 
at  right  angles  to  the  magnetic  meridian,  it  will  also  be  vertical 
when  the  circle  M  is  turned  through  an  angle  of  180°  from 
this  position.  Two  points  on  this  circle  must  therefore  be 
found  such  that  the  needle  points  towards  them  when  it  is  in 
one  position,  and  also  when  the  same  circle  is  given  a  rotation 
of  1 80°.  This  will  determine  at  the  same  time  both  the  mag- 
netic meridian  and  the  two  points  in  the  vertical  diameter  of 
the  graduated  circle.  The  horizontal  diameter  can  then  be 
inferred. 

II.  Readings  should  be  taken  at  both  ends  of  the  needle  in 
order  to  correct  any  error  arising  from  the  axis  of  rotation  not 
passing  exactly  through  the  center  of  the  vertical  circle. 

III.  The  magnetic  axis  of  the  needle  may  not  coincide  with 
its    geometrical    axis,  and  to  correct   errors    from  this  source, 
the  method  of  reversion   indicated    in  Experiment  III.  should 
be  adopted.     Instead  of  removing  the  needle,  this  can  be  ac- 
complished by  simply  rotating  the  vertical  circle  through   180° 
when  any  reading   has  been    taken,  and    again    observing   the 
inclination. 

IV.  The  center  of  inertia  may  not  lie  in  the  axis  of  suspen- 
sion.    In  this  case,  the  inclination  will  be  affected  by  gravity,  and 


HORIZONTAL   INTENSITY. 


195 


should  be  corrected  by  repeating  the  operation  after  the  needle 
has  been  remagnetized  so  that  its  poles  are  reversed.  As  in 
the  case  of  declination,  maps  are  used  shewing  the  inclination 
at  various  places.  The  lines  which  pass  through  those  points 
where  the  inclination  is  the  same  are  termed  isoclinic. 

V.    DETERMINATION    OF    THE    ABSOLUTE    INTENSITY 
OF   THE    EARTH'S    MAGNETIC   FIELD. 

In  Experiment  IV.  it  has  been  shewn  that  if  the  dipping  needle 
be  slightly  displaced  from  its  position  of  equilibrium  in  the  mag- 
netic meridian,  it  will  perform  small  oscillations  whose  periodic 

time  is  2  TT'Y— L,  M-^  being  the  moment  of  inertia  of  the  needle 

about  the  axis  of  rotation,  M  its  magnetic  moment,  and  F  the 
resultant  intensity  of  the  earth's  magnetic  action.  It  follows 

from  this  that  MF= ^ — -,  and  since  Ml  and  /  can  be  easily 

ascertained,  a  relation  can  thus  be  established  between  M 
and  F.  Owing  to  friction,  however,  an  accurate  result  cannot 
be  determined  in  this  way,  and  it  is  customary  to  determine 
instead  the  product  MH,  H  being  the  horizontal  component 

J[/T 

of  F,  and  then  by  means  of  a  second  relation  — ,  to  evaluate 

H 
both  the  quantities  H  and  M. 


Fig.  7.  Fig.  8. 

After  H  has  been  found  in  this  manner,  and  the  inclination 
determined  at  the  point  of  observation  by  a  separate  investiga- 
tion, the  resultant  intensity  F  can  then  be  calculated  from  the 
relation  F=H-secD. 


EXPERIMENTAL   PHYSICS. 


The  apparatus  employed  in  this  experiment  consists  of  a 
telescope  (Fig.  7)  mounted  so  as  to  be  capable  of  rotation  in 
a  horizontal  and  a  vertical  plane,  a  finely  divided  scale  VV 
(Fig.  8),  and  a  magnetometer. 

This  instrument  (Fig.  9)  is  composed  essentially  of  a  cylin- 
drical brass  or  copper  box  B  resting  on  a  tripod  provided  with 
leveling  screws.  It  is  ca- 
pable of  rotation  about  a 
vertical  axis,  carries  with  it 
a  graduated  circle  C  at- 
tached to  its  base,  and  has 
its  anterior  face  pierced  by 
a  circular  opening  in  which 
is  inserted  a  convergent 
lens  O  whose  focal  length 
is  about  one  meter.  A 
metallic  column  V  has  its 
base  inserted  in  a  beveled 
plate  resting  on  the  top  of 
B,  and  carries  at  its  upper 
end  a  reel  on  which  is 
wound  the  silk  strand  which 
supports  the  stirrup  and 
the  bar  magnet  A.  This 
stirrup  carries  a  vertical 
mirror  M  mounted  so  as 
to  be  at  right  angles  to 

the  magnet  when  the  latter  is  in  position.  A  second  mirror 
M'  is  fitted  into  a  frame  fastened  to  the  bottom  of  the  box, 
and  can  be  adjusted  either  horizontally  or  vertically  by  means 
of  the  screws  E,  E.  A  divided  circle  C  indicates  any  rotation 
given  to  the  column  V. 

Theory.  — Determination  of  MH.  When  a  bar  magnet  is  sus- 
pended so  that  it  is  free  to  move  in  a  horizontal  plane,  it  will, 


Fig.  9. 


HORIZONTAL  INTENSITY. 


197 


when  displaced,  oscillate  about  its  mean  position.  If  H  is  the 
horizontal  intensity  of  the  terrestrial  field,  and  M  the  magnetic 
moment  of  the  bar,  the  equation  of  motion  is 


The  time  of  a  small  oscillation  is  therefore  given  by 


and  denoting  the  moment  of  inertia  of  the  magnet  by  Mv  this 
equation  gives  the  relation 

(2) 


The  quantity  t  can  easily  be  ascertained  by  counting  the 
oscillations  performed  in  some  given  time,  and  M-^  can  either 
be  calculated  from  the  dimensions  of  the  magnet  or  found 
experimentally. 

In  the  case  of  very  exact  determinations  allowance  will  have 
to  be  made  for  the  torsion  couple  of  the  suspending  fiber. 
Adopting  this  correction,  the  relation  becomes 

(3) 

the  constant  C  being  the  coefficient  of  torsion,  which  must  be 
ascertained  by  a  preliminary  investigation.  (See  Appendix  B.) 

Determination  of  —  --  The  relation-—  is  determined  from 
H  H 

the  reciprocal  actions  exerted  by  two  magnets  situated  at  a 
distance  from  each  other,  which  is  great  compared  with  their 
lengths.  After  having  accurately  determined  the  time  of  vibra- 
tion of  the  magnet  A,  it  is  next  used  to  deflect  a  second  one 
which  is  inserted  in  the  stirrup  in  its  place. 

Either  of  two  methods  may  be  followed.  In  one  the  disposi- 
tion is  that  shewn  in  Fig.  10,  where  JVS  represents  the  magnet 


198  EXPERIMENTAL   PHYSICS. 

in  the  stirrup  at  rest  in  the  meridian,  and  AB  is  the  magnet  A 
resting  on  a  support  in  the  same  horizontal  plane  as  NS,  but 
pointing  east  and  west.  In  the  second,  the  arrangement  is 


# 

1 


Fig.  10. 


shewn  in  Fig.  12.  The  magnet  NS  is  suspended  in  the 
meridian  as  before,  but  the  deflecting  magnet  AB  is  now  placed 
with  its  center  in  this  plane  and  its  axis  at  right  angles  to  it. 

METHOD  I.  —  Denoting  the  lengths  of  AB  and  NS  (Fig.  10) 
by  /  and  /v  respectively,  and  the  strengths  of  their  poles  by 


Fig.   11. 

fi  and  /*!,  it  follows  if  r,  the  distance  between  their  centers,  is 
considerable,  that  the  action  exerted  by  AB  on  each  of  the 
poles  of  NS  is  approximately  equal  to 


HORIZONTAL   INTENSITY. 


199 


and  may  be  taken  as  acting  parallel  to  the  line  joining  the 
centers  of  the  magnets.  Under  these  actions  and  those  exerted 
by  the  earth's  field,  the  magnet  NS  will  take  up  the  position 
indicated  in  Fig.  u.  Taking  moments  about  the  point  of 
suspension,  it  follows  that 


--  r 

2 


from  which  r3tan  6  =  2  —fi  +-^A  (4) 

If  now  this  experiment  be  repeated  with  the  centers  of  the 
magnets  at  a  distance  rlt  apart,  we  will  have  a  second  relation, 


and  —  is  therefore  given  by  the  equation, 
H 


M  _  —  .„ 

—  ~ 


METHOD  II.  —  Denoting  again  the  distance  between  the  cen- 
ters of  the  magnets  by  r,  and  their  lengths  by  /  and  /1}  the 


S 


Fig.   12. 


actions  exerted  by  AB  on  5  and  N  respectively  (Fig.  12)  are 
perpendicular  to  the  meridian,  and  are  approximately  equal  to 


2QO  EXPERIMENTAL   PHYSICS. 

Under  these  forces,  and  those  exerted  by  the  earth's  field,  the 
suspended  magnet  will  again  be  deflected  from  the  meridian 
(neglecting  any  small  motion  of  translation)  through  an  angle 
6,  the  equation  of  moments  being,  in  this  case, 

(7) 


From  this  it  follows  that 


and  repeating  the  experiment,  taking  the  distance  between  the 
centers  to  be  rlt  we  have  also 

Combining  (8)  and  (9),   a  second  equation  for  determining 
is  given  by  Jf 


H  r*—r^ 

Experiment.  —  When  the  apparatus  is  arranged  for  taking 
observations,  the  several  parts  are  situated  as  in  Fig.  13,  where 
AB  represents  the  meridian,  CA  the  telescope,  VV  the  scale, 
and  GH  and  KL  the  mirrors  attached  to  the  base  of  the 
magnetometer  and  to  the  stirrup  respectively.  The  first  step 
to  take  in  making  this  disposition  is  to  adjust  the  mirror  GH, 
so  that  it  is  perpendicular  to  the  meridian.  This  is  arrived  at  by 
placing  a  copper  bar  in  the  stirrup,  and,  after  taking  all  the 
torsion  out  of  the  silk  strand,  by  turning  the  column  V  until 
the  bar  lies  approximately  in  the  meridian,  as  indicated  by  a 
delicately  poised  compass  needle.  On  replacing  the  copper  bar 
by  the  magnet,  the  latter  will  then  come  to  rest  in  the  meridian 
without  being  subjected  to  any  torsion,  and  the  mirror  KL  will 
therefore  be  perpendicular  to  it.  The  magnetometer  is  next 
turned  until  the  mirror  GH  is  parallel  to  KL.  This  can  be 


HORIZONTAL  INTENSITY. 


2O I 


performed  very  accurately  by  observing  with  the  telescope  the 
images  of  the  scale  (GH  being  slightly  inclined  to  the  vertical) 
formed  by  the  two  mirrors,  the  latter  being  parallel  when  the 
divisions  on  one  scale  image  are  seen  directly  above  the  corre- 
sponding ones  on  the  other. 

In  order  to  adjust  the  telescope  so  that  its  axis  will  be  in 
the  meridian,  place  the  scale  with  its  stand  on  the  same  sup- 
port as  the  former,  and  then,  after  noting  the  scale  division, 


generally  zero,  which  is  situated  in  the  same  plane  as  the 
axis  of  the  telescope  and  the  vertical  diameter  of  its  objective, 
move  the  support  until  the  images  of  this  division,  formed 
by  the  mirrors,  are  seen  on  the  vertical  cross-hair  in  the  eye- 
piece. The  axis  of  the  telescope  will  then  be  in  the  meridian, 
and  the  scale  at  right  angles  to  it.  In  determining  the  quantity 
MH,  it  is  first  necessary  to  find  accurately  the  time  of  a  single 
oscillation.  This  is  done  by  giving  the  magnet  A  initially  a 
small  displacement,  by  bringing  near  to  it  a  piece  of  iron  or 
steel,  and  when  the  motion  has  become  steady,  by  ascertaining 
with  a  stop  watch  the  time  occupied  in  performing  a  large 


202  EXPERIMENTAL   PHYSICS. 

number  (one  hundred,  for  example)  of  small  oscillations.  The 
time  of  a  single  one  can  then  be  calculated,  and  by  taking  the 
mean  of  the  results  obtained  by  repeating  this  operation  a 
number  of  times  a  very  close  approximation  can  be  found.  In 
order  to  make  certain  that  the  stop  watch  marks  correct  time 
it  should  be  compared,  both  before  and  after  the  experiment, 
with  a  standard  chronometer. 

As  already  indicated,  the  moment  of  inertia  of  the  magnet 
A  and  the  stirrup  upon  which  it  rests  may  be  determined 
experimentally  by  the  method  indicated  in  Appendix  B  ;  but 
for  a  rough  approximation  that  of  the  stirrup  may  be  neglected, 
and  that  of  the  magnet  calculated  from  its  dimensions. 

In  determining  — ,  a  second  magnet  is  placed  in  the  stirrup, 
H 

and  the  magnet  A  rests  on  a  support  attached  to  a  copper 
bar  (not  shewn  in  the  figure),  which  together  with  a  counter- 
poise rotates  about  the  cylindrical  column  connecting  the  box 
of  the  magnetometer  to  the  tripod.  Divisions  are  marked  on 
this  bar,  shewing  the  distance  of  the  center  of  the  support 
from  the  axis  of  rotation  of  the  instrument,  and  therefore  that 
between  the  centers  of  the  magnets.  The  mirror  GH  is  gen- 
erally attached  to  the  base  of  the  magnetometer  so  that  its 
lower  edge  lies  in  the  ninety-diameter  of  the  divided  circle  C, 
and  therefore  when  the  bar  carrying  A  is  turned  so  that  its 
index  points  to  this  division  it  is  perpendicular  to  the  meridian, 

and  —  may  then  be  found  by  Method  I. 
H 

If  it  is  desired  to  adopt  Method  II.,  the  bar  should  be  turned 
through  a  right  angle  from  this  position. 

The  method  of  Poggendorff  is  followed  in  measuring  the 
angle  through  which  the  auxiliary  magnet  is  deflected.  When 
this  magnet  is  in  the  meridian,  one  sees  the  image  of  the 
zero  division  (Fig.  13)  formed  on  the  cross-hair  of  the  telescope, 
but  when  the  mirror  KL  is  turned  through  an  angle  6,  one 
sees  formed  there  the  image  of  some  other  division,  N. 


HORIZONTAL   INTENSITY. 


203 


It  follows  then  that 


tan  26  = 


AN 
AS 


1  AN 

or  0=-.——  approximately. 

2  *LJJ 

In  ascertaining  the  value  of  6  corresponding  to  a  given  value 
of  r  the  magnet  A  should  first  be  placed  with  its  positive  pole 


Fig.  14. 


pointing  east,  for  example,  and  the  deflection  noted,  and  it 
should  then  be  turned  with  its  negative  pole  in  this  direction, 
and  the  deflection  again  observed.  After  this  the  bar  should 


204 


EXPERIMENTAL   PHYSICS. 


be  rotated  through  180°  and  the  same  process  repeated,  the 
mean  of  the  four  values  so  obtained  being  the  one  inserted  in 
the  formula  with  the  corresponding  value  of  r. 

The  apparatus  is  designed  specially  for  the  determination  of 
the  quantity  H,  and  is  not  suited  for  finding  the  magnetic 
moments  of  magnets  of  different  sizes. 

When,  however,  H  has  been  determined  accurately  for  the 
point  of  observation,  these  can  readily  be  obtained  with  a 
slightly  modified  form  of  the  instrument,  by  simply  performing 
the  first  operation  in  this  experiment. 


Fig.  15. 

In  the  British  magnetic  observatories,  and  also,  in  fact,  in 
many  laboratories,  the  Kew  portable  magnetometer  is  used  for 
finding  the  horizontal  intensity. 

In  Fig.  14  it  is  shewn  adjusted  for  determining  the  quantity 
MH.  The  vibration  magnet  M±  is  suspended  at  the  center  of 
a  wooden  box  by  a  silk  thread  attached  to  the  torsion  head  A. 
The  magnet  is  a  hollow  steel  cylinder  having  a  glass  scale  at 


HORIZONTAL    INTENSITY.  205 

one  end  and  a  lens  at  the  other,  the  former  being  in  the  focal 
plane  of  the  lens.  This  scale  is  illuminated  by  a  small  mirror, 
C,  and  the  oscillations  of  the  magnet  are  observed  by  means 
of  the  telescope  Tv 

The  moment  of  inertia  of  the  magnet  is  found  experiment- 
ally, by  inserting  a  cylindrical  brass  or  copper  bar  whose 
moment  of  inertia  can  be  calculated,  in  the  brass  collar  which 
is  shewn  immediately  above  the  magnet. 

The  arrangement  adopted  in  determining  —  is  exhibited  in 

H 

Fig.  15.  The  wooden  box  is  removed  and  a  metallic  one  put 
in  its  place.  The  vibration  magnet  Ml  is  placed  on  the  support 
C,  which  is  movable  along  the  bar  SS'  previously  referred  to, 
and  the  telescope  T,  with  its  accompanying  scale  s,  is  attached 
to  the  lower  part  of  the  end  of  the  large  hollow  cylinder. 
The  auxiliary  magnet,  Mz,  and  the  mirror  for  reflecting  the 
scale  divisions  are  shewn  in  position  in  the  figure. 

In  determining  H  by  this  instrument  the  same  theory  and 
the  same  general  method  of  manipulation  as  that  just  described 
apply. 

VI.    MAGNETIC   FIELD   OF   A   CURRENT.  —  BIOT   AND 
SAVART'S    LAW. 

Oerstedt  discovered  in  1820  that  a  magnetic  needle,  if  sus- 
pended in  the  neighbourhood  of  a  rectilinear  current,  tended  to 
place  itself  at  right  angles  to  the  conducting  wire.  This  dis- 
covery at  once  suggested  that  surrounding  a  rectilinear  current 
there  is  a  magnetic  field  and  that  its  lines  of  force  are  a  series 
of  concentric  circles  having  the  conductor  as  a  common  axis. 
The  truth  of  this  conclusion  has  been  amply  verified  by  experi- 
ment and  can  be  clearly  exhibited  by  the  use  of  iron  filings. 

If  a  straight  wire  in  which  a  current  is  established  be  passed 
up  through  a  piece  of  cardboard  which  is  held  horizontally,  and 
fine  iron  filings  be  then  sprinkled  upon  it,  these  will  at  once 
arrange  themselves  into  the  series  of  circles  described  ;  while,  on 


206 


EXPERIMENTAL   PHYSICS. 


the  other  hand,  if  the  wire  be  laid  flat  on  the  paper,  the  filings 
will  arrange  themselves  into  a  series  of  straight  lines  cutting 
the  line  of  the  current  at  right  angles. 

Since  then  the  lines  of  force  are  circles,  it  only  remains  to 
state  the  sense  in  which  a  north  magnetic  pole  tends  to  move, 
in  order  to  define  completely  the  action  exerted  by  a  given  cur- 
rent on  a  magnet.  The  following  rule  has  been  proposed  by 
Ampere  for  guidance  in  this  connection  :  The  positive  pole  of 
a  magnet  tends  to  turn  towards  the  left  hand  of  an  observer  look- 
ing at  it,  and  supposed  swimming  in  the  conductor  in  the  direc- 
tion in  which  it  has  been  agreed  to  consider  the  current  as  passing. 

A.    The  magnetic  field  of  an  infinitely  long  rectilinear  current. 
As  in  the  case  of  other  magnetic  fields  the  intensity  of  that 
produced  by  a  long  rectilinear  current  may  be  investigated  by 
the  method  of  oscillations.     The  current  is  passed  through  a 
vertical  conductor  as  in  Fig.  16,  and 
a  small  magnet  is  suspended  by  a 
torsionless  fiber  at  different  points 
along    a    horizontal     line     passing 
through  the  wire  perpendicularly  to 
the  direction  of  the  magnetic  meri- 
dian at  that  point. 

If  the  needle  is  taken  very  small 
and  is  placed  not  too  close  to  the 
wire,  the  action  of  the  current  on  it 
will  reduce  to  that  of  a  couple  which 
acts  in  conjunction  with,  or  in  opposition  to,  the  horizontal 
component  of  the  earth's  magnetic  field  according  to  the  direc- 
tion in  which  the  current  passes. 

By  applying  this  method  Biot  and  Savart  found  that  when  a 

constant  current  was  maintained  in  the  wire  the  intensity  of  the 

field  was  inversely  proportional  to  the  distance  from  the  conductor. 

Denoting  the  horizontal  component  of  the  earth's  action  near 

the  current  by  H,  and  the  intensity  of  the  field  due  to  the  cur- 


Fig.  16. 


MAGNETIC   FIELD    OF   A   CURRENT. 


207 


rent  at  the  distances  a  and  b  by  F^  and  F2  respectively,  the 
equations  of  motion  for  the  magnet  when  acted  on  by  (i)  H 
alone,  (2)  Fl  and  H  together,  and  (3)  Fz  and  H  together,  are 

(i) 
,  (2) 

and  Sair8^  =  -  (ff+FJMsm  0.  (3) 

If  «,  »lf  and  #2  are  the  number  of  oscillations  performed  in 
each  of  these  cases  respectively  in  a  given  time,  we  have,  if  the 
amplitudes  of  the  vibrations  are  small, 

(4) 
(5) 
(6) 


where  K  is  a  constant  depending  on  the  moment  of  inertia,  and 
the  magnetic  moment  of  the  magnet. 
From  (4),  (5),  and  (6)  it  follows  that 


7^~«22-«2 

If  then  the  law  of  the  inverse  distance  holds,  we  should  have 

FI     b  r        b     «n2 —  7z2 

—=-,  and  therefore  -=-L -,   i.e.  we  should  have  b(ns— «2) 

F2     a  a     nz2-nz 

=a(nlz—nz)=  a  constant  for  all  positions  of  the  magnet  along  a 
line  through  the  conductor  perpendicular  to  the  meridian. 

The  apparatus  for  performing  this  experiment  is  similar  to 
that  shewn  in  Fig.  9.  The  various  parts  are  arranged  just  as 
indicated  there  excepting  the  wire  conductor,  which  is  movable 
about  the  vertical  through  the  point  of  suspension  of  the 
magnet,  and  is  so  placed  that  the  perpendicular  on  it  from 
the  magnet  is  at  right  angles  to  the  meridian. 


208 


EXPERIMENTAL   PHYSICS. 


In  order  to  make  this  adjustment  accurately  the  wire  is  first 
placed  approximately  in  position,  and  the  magnet  is  brought  to 
rest  in  the  meridian.  If  the  current  is  then  turned  on  and  the 
magnet  remains  still  at  rest  the  adjustment  is  perfect ;  if,  how- 
ever, there  is  a  displacement  of  the  magnet  the  wire  must  be 
rotated  through  a  small  angle  in  a  sense  opposite  to  that  of  such 
displacement  and  the  same  test  again  applied. 

As  the  chief  difficulty  in  this  experiment  arises  from  the 
inconstancy  of  the  current,  a  sensitive  galvanometer  should  be 
inserted  in  the  circuit  along  with  an  adjustable  resistance,  so 
that  any  variation  in  the  current  strength  may  be  at  once 
detected,  and  corrected  by  altering  the  resistance.  A  battery 
of  Grove  cells  will  be  found  to  produce  a  very  steady  current 
for  a  considerable  time. 

In  order  to  correct  still  further  any  variations  in  the  current, 
the  oscillations  of  the  magnet  should  be  counted  a  number  of 
times  for  each  position. 

B.   Magnetic  field  produced  by  an  angular  current. 

Biot  and  Savart  modified  the  experiment  just  described  by 

observing  the  action  of  a  cur- 
rent traversing  two  very  long 
rectilinear  conductors  BA  and 
AC,  so  connected  as  to  make 
an  angle  at  A,  Fig.  17,  upon 
a  short  magnet  placed  with  its 
center  P  on  the  line  bisecting 

the  angle  BAG. 

As  can  be  seen  from  the 
figure,  the  circular  lines  of 
force  due  to  the  current  pass- 
ing along  BA  and  AC  have  a 
common  tangent  at  P,  and  if 
these  wires  are  so  placed  that 
their  plane  is  vertical  and  per- 


MAGNETIC    FIELD    OF   A   CURRENT. 


209 


pendicular  to  the  magnetic  meridian  through  A,  the  action 
exerted  on  a  short  magnet  suspended  horizontally  at  P  will 
again  be  reduced  to  two  equal  and  opposite  forces  whose  direc- 
tion is  in  the  meridian  at  that  point. 

The  results  of  experiment  indicate  that  the  action  of  such 
a  disposition  of  conductors  is  given  by 


where  A  is  the  angle  BAP,  and  r  is  the  length  AP,  K  being  a 
constant. 

The  same  method  is  adopted  in  verifying  this  law  as  in  that 
of  the  inverse  distance  for  an  infinite  rectilinear  current,  and 
similar  precautions  are  necessary  regarding  the  constancy  of 
the  current,  and  the  plane  BAG  being  perpendicular  to  the 
meridian. 


Law  of  Laplace. 

It  follows  from  the  law  just  investigated  that  the  action  of 
an  infinite  current,  supposed  commencing  at  A  (Fig.  18),  upon 
a  magnetic  pole  placed  at  P  can  be  represented  by 


2        r 
where  BAP  is  the  angle  A,  and  AP  is  the  length  r. 


2io  EXPERIMENTAL   PHYSICS. 

It  follows,  therefore,  that 


=  K- 


P 

where  p=PN,  and  since  the  action  of  AB  on  P  may  be  con- 
sidered as  the  sum  of  the  elementary  actions  of  its  parts,  that 
of  an  element  A  A'  is  given  by 

sin  d  cos  - 
dF=K  -  -  -  -dA 


dA 
i.e.  since 

AA 

and  p=r  sin  A, 


or,  denoting  the  length  AA' by  ds,  we  have  Laplace's  law  for 
the  action  of  an  element  of  current, 


VII.     MEASUREMENT   OF  CURRENT   INTENSITY. 

Of  the  many  effects  that  can  be  produced  by  an  electric  cur- 
rent, two  have  been  selected  as  being  most  suitable  for  the 
determination  of  its  intensity, — the  action  exerted  by  the 
current  on  a  magnetic  pole,  and  the  decomposition  of  chemical 
compounds. 

If  a  circuit  be  so  arranged  that  a  constant  current  in  pass- 
ing can  produce  (i)  magnetic,  (2)  heating,  (3)  chemical,  and 
(4)  magnetizing  effects,  it  will  be  found  that  if  a  measure  of 
each  of  these  be  taken,  only  those  of  the  magnetic  and  the 
chemical  effects  will  bear  the  same  ratio  to  the  new  values 
obtained  when  the  current  is  by  any  means  altered  in  intensity. 


MEASUREMENT   OF   CURRENT   INTENSITY. 


211 


It  has  therefore  been  agreed  to  consider  current  strength  as 
being  proportional  to  the  action  in  each  of  these  cases  ;  and 
vice  versa,  the  action  exerted  on  a  magnetic  pole,  and  the 
amount  of  chemical  separation  produced,  will  therefore  be  pro- 
portional to  the  intensity  of  the  current  causing  such  action. 

If,  then,  a  magnetic  pole  of  strength  p  be  placed  near  a 
conductor  bearing  a  current  whose  intensity  is  C,  Laplace's 
law,  when  combined  with  the  above  convention,  gives 

.  „     r_  Cds  sin  A 


as  the  force  exerted  on  the  pole  by  an  element  of  current  ds, 
A  being  the  angle  between  the  line  r  and  the  element  ds. 

The  absolute  unit  of  current  strength  in  the  electro-magnetic 
system  is  defined  to  be  such  that  if  one  centimeter  length  of 
the  circuit  be  bent  into  an  arc  of  one  centimeter  radius,  the 
current  in  it  exerts  a  force  of  unit  intensity  upon  a  magnetic 
pole  of  unit  strength  placed  at  the  center  of  the  arc.  On  the 

basis  of  this  definition,  therefore,  /* — s-^- —  will  represent  the 

force  in  absolute  units  exerted  on  the  magnetic  pole  /j,  placed 
as  previously  indicated. 


Fig.   19. 


Action  of  a  circular  current  on  a  magnetic  pole. 
Let  AC  represent  the  circular  conductor  bearing  the  current 
whose   intensity  is  C  (Fig.   19),  AB  or  ds  an  element  of  this 


212  EXPERIMENTAL   PHYSICS. 

current,  and  P  the  position  of  a  magnetic  pole  of  strength  p., 
supposed  placed  on  CD,  the  axis  of  the  circuit. 

Since  in  this  case  the  angle  A  is  a  right  angle,  the  force 

exerted  by  the  element  ds  on  the  pole  p is  equal  to  ^  2S.    It  also 

acts  in  a  direction  at  right  angles  to  APB.     If  a  is  the  radius  of 
the  circle  AC,  and  PD  be  denoted  by  x,  the  component  of  this 

force  parallel  to   CD   is   equal   to  **'    '  s'a       Since    a   similar 

(02+*2)* 
expression  may  be  obtained  for  each  element  of  the  circuit,  the 

action  of  the  whole  conductor  resolved  along  CD  is  given  by 


and  from  the  symmetry  of  the  arrangement  it  is  evident,  more- 
over, that  this  is  the  resultant  action  of  the  current. 

If  instead  of  a  single  coil  there  are  n  of  them,  the  action  will 
be  given  by 

*-*:&$  (2> 

and  if  the  pole  is  situated  at  the  center  of  the  coil,  this  further 
reduces  to 

F=**!f*.  (3) 

The  tangent  galvanometer. 

A  form  of  the  tangent  galvanometer  commonly  used  is  that 
shewn  in  Fig.  20.  It  consists  of  a  coil  of  insulated  wire  wound 
on  the  vertical  circle  AB,  which  is  capable  of  rotation  about  an 
axis  coinciding  with  its  vertical  diameter.  At  the  center  of  this 
circle  a  short  magnet  is  suspended  from  C  by  a  torsionless 
thread  of  silk,  and  immediately  below  it  there  is  placed  a  hori- 
zontal divided  circle  DE,  which  can  rotate  about  the  same 
axis  as  AB,  but  independently  of  it.  The  divided  circle  FG, 
which  is  provided  with  a  vernier,  indicates  the  amount  of  any 


MEASUREMENT   OF   CURRENT   INTENSITY. 


213 


rotation  given  to  the  coil  bearing  the  current,  and  an  alumin- 
ium pointer  attached  to  the  magnet  serves  to  shew  the  angle 
through  which  the  latter  may  be  deflected. 

In  adjusting  the  instrument 
for  a  measurement,  the  circles 
DE  and  FG  are  first  made 
horizontal  by  means  of  the 
leveling  screws,  and  the  coil 
AB  is  then  turned  until  the 
magnet  when  at  rest  lies  ap- 
parently in  its  plane,  which 
will  then  coincide  approxi- 
mately with  the  magnetic 
meridian. 

Since  the  magnet  is  short, 
the  actions  exerted  on  its  poles 
by  the  current  will  form  a 
couple  which  will  tend  to 
make  it  take  up  a  position  at 
right  angles  to  the  meridian. 
Owing,  however,  to  the  pres- 
ence of  the  terrestrial  field,  it 
will  come  to  rest  in  a  posi- 
tion such  as  that  indicated  in 

Fig.  21,  where  AB  is  the  meridian,  CD  the  magnet,  and  6  the 
angle  of  deviation. 

When,  therefore,  the  current  is  passing  and  the  magnet  has 
come  to  rest,  this  angle  0  should  be  noted ;  and  if,  when  the 
current  is  reversed,  the  magnet  is  deflected  through  the  same 
angle  but  on  the  other  side  of  the  meridian,  the  plane  of  the 
coil  coincides  with  the  latter,  and  the  instrument  is  ready  for  a 
measurement.  If,  however,  this  condition  does  not  obtain,  the 
coil  AB  must  be  given  a  slight  displacement,  and  the  test 
again  applied. 


214 


EXPERIMENTAL   PHYSICS. 


When  the  magnet  is  displaced,  as  in  Fig.  21,  it  follows,  by 
taking  moments  about  the  point  of  suspension  that, 


(4) 


.Ha 
2  HIT 


tan0, 


and  since  the  right-hand  member  of  this  equation  involves  quan- 
tities which   can  be  determined  with   precision,  this   method 


affords  a  means  of  accurately  measuring  the  intensity  of  a  cur- 
rent in  absolute  units. 

The  ampere,  or  practical  unit  of  current  strength,  is  equal  to 
one-tenth  of  the  absolute  unit.  The  relation  for  determining 
the  intensity  of  a  given  current  in  amperes  is,  therefore, 

.     ZL7"_ 

(5) 


To  obtain  satisfactory  results,  H  must  be  found  by  means 
of  the  magnetometer  at  the  point  where  the  galvanometer  is 
placed,  and  extreme  care  must  be  taken  to  have  the  coil  AB 
accurately  in  the  plane  of  the  meridian.  In  order  to  overcome 
any  imperfection  in  this  adjustment,  the  current  should  always 
be  reversed  and  the  mean  of  the  two  readings  taken. 


MEASUREMENT   OF   CURRENT   INTENSITY. 


215 


For  very  close  determinations,  it  is  best  to  have  only  one  turn 
of  wire  in  the  circuit,  as  in  that  case  the  radius  a  can  then  be 
measured  very  accurately. 

In  order  to  dampen  the  oscillations  of  the  magnet,  a  plate  of 
copper  is  frequently  placed  immediately  below  it  within  the 
circle  DE;  and,  as  it  generally  takes  a  considerable  time  for 
the  magnet  to  cease  vibrating,  even  when  this  precaution  is 
taken,  its  position  of  equilibrium  is  generally  deduced  by  noting 
three  of  its  successive  positions  of  rest. 

In  many  cases  the  angle  of  deflection  is  determined  by  means 
of  a  small  mirror  attached  to  the  magnet.  A  scale  and  telescope 
may  then  be  used  just  as  in  Experiment  V.,  and  the  angle 
through  which  the  needle  is  deflected  calculated  by  the  method 
there  explained. 

Besides  being  used  for  measuring  given  currents,  the  tangent 
galvanometer  is  frequently  employed  to  test  the  accuracy  of 
direct  reading  ammeters.  These  again,  when  they  have  been 
accurately  calibrated,  may  be  used  to  calculate  the  constant  K 
in  the  formula  C=K  tan  0,  for  a  tangent  galvanometer  whose 
construction  is  of  such  a  character  that  it  is  difficult  to  measure 
with  accuracy  the  dimensions  of  its  coil. 

Sine  galvanometer. 

The  instrument  just  described  may  also  be  used  as  a  sine 
galvanometer.  _  The  same  precautions  are  to  be  taken  in  regard 
to  placing  the  coil  AB  initially  in  the  meridian,  and  the  same 
adjustments  made,  except  that  when  the  current  is  turned  on, 
and  the  magnet  is  deflected,  the  coil  AB  is  then  slowly  and 
carefully  rotated  until  it  overtakes  the  magnet,  which  when 
it  comes  to  rest  does  so  with  its  axis  in  the  plane  of  the  coil. 

Figure  22  exhibits  the  forces  which  then  maintain  the  magnet 
in  equilibrium,  AB  denoting  the  meridian.  By  again  taking  mo- 
ments about  the  point  of  suspension,  we  obtain  the  relation, 

C  =  ^Lsm0,  (6) 

2;/?r 


216 


EXPERIMENTAL   PHYSICS. 


and  from  it  the  intensity  of  any  current  can  be  calculated. 
The  angle  6,  which  is  that  through  which  the  coil  is  rotated, 
can  be  ascertained  by  means  of  the  graduations  on  FG.  Es- 
pecial care  should  be  taken  to  have  the  pointer  attached  to  the 
magnet  directed  in  its  initial  and  final  positions  to  the  same 
division  on  the  graduated  circle  DE,  which  rotates  with  the  coil. 


Fig.  22. 


VIII.     HYDROGEN   VOLTAMETER. 

When  a  current  of  electricity  is  made  to  decompose  an  elec- 
trolyte it  is  found  that  the  amount  of  a  gas  liberated,  or  the 
quantity  of  a  metal  deposited,  is  directly  proportional  to  the 
intensity  of  the  current,  and,  within  very  wide  limits,  is  not 
affected  by  the  size  or  shape  of  the  vessel  containing  the 
electrolyte. 

The  mass  of  a  substance  liberated  in  one  second  by  one 
ampere  of  current  is  termed  its  electro-chemical  equivalent. 
When,  therefore,  a  constant  current  is  passed  through  a  voltam- 
eter and  a  known  weight  of  gas  is  liberated  in  a  second,  the 
strength  of  this  current  is  perfectly  defined,  and  can  readily  be 
calculated  when  the  electro-chemical  equivalent  of  the  gas  is 
known.  That  of  hydrogen  is  .00x301038  grams. 


HYDROGEN    VOLTAMETER. 


217 


A  simple  form  of  hydrogen  voltameter  is  shewn  in  Fig.  23. 
The  acidulated  water  is  placed  in  the  vase  AB,  whose  base  is 
pierced  by  two  holes  in  which  are  cemented  two  small  strips  of 
platinum.  These  are  connected  to  the  two  binding  poles  C 
and  D,  at  which  the  current  is  made  to  enter  and  leave  the 


Fig.  23. 

voltameter.  The  hydrogen  is  collected  in  the  graduated  glass 
tube  E,  and  the  current  is  made  to  pass  through  the  mercury 
in  the  small  cup  F,  so  that  the  circuit  can  be  rapidly  made  or 
broken. 

When  the  current  is  established,  the  oxygen  which  appears 
at  the  positive  platinum  partly  recombines  with  the  water,  and 
since  the  quantity  of  it  which  might  be  collected  would  not 
then  be  an  exact  measure  of  the  decomposition  which  has  taken 
place,  it  is  customary  to  allow  it  to  escape  and  to  collect  only 
the  hydrogen,  by  placing  the  tube  E  over  the  negative  platinum. 

When  an  experiment  is  about  to  be  performed  the  circuit  is 
first  broken  at  F,  and  the  tube  E  is  then  carefully  filled  with 
the  acidulated  water  and  inverted  over  the  negative  electrode. 
A  thermometer  is  also  placed  in  the  liquid  in  the  vase. 

The  circuit  is  then  closed  by  inserting  the  leading  wire  in 
the  mercury,  and  a  stop  watch,  reading  to  one-fifth  of  a  second, 


2Ig  EXPERIMENTAL   PHYSICS. 

is  started  at  the  same  instant.  When  the  tube  has  been  filled 
with  hydrogen  until  the  liquid  in  it  stands  at  the  same  level 
as  that  in  the  vase  outside,  the  circuit  is  broken  and  the  watch 
stopped.  The  time,  /,  during  which  the  current  was  passing 
can  thus  be  accurately  found,  and  the  volume,  V,  in  cubic  centi- 
meters, occupied  by  the  hydrogen  read  directly  from  the  gradu- 
ations on  the  tube.  The  temperature  of  the  gas,  T.°  C,  may  be 
taken  to  be  the  same  as  that  of  the  liquid  in  the  vase.  The 
pressure,  f,  exerted  by  the  water  vapour  corresponding  to  this 
temperature  can  be  found  from  the  tables,  and  if  H  is  the 
reduced  height  of  the  barometer,  the  pressure  to  which  the 
hydrogen  is  subjected  is  proportional  to  (//"—/). 

Since  the  weight  of  I  c.c.  of  dry  air  at  760  mm.  pressure  and 
o°  C.  is  .001293  grams,  and  the  density  of  hydrogen  is  .06926, 
the  weight  of  the  gas  liberated  is  given  by 

W=  Vx  -OP  1  293  x  .06926  x  (H-f)  (  . 

760(1  +.003665  T) 

The  average  intensity  of  the  current  in  amperes  is  therefore 
found  from  the  relation 


c=  .-  , 

/x  760(1  +.003665  T) 

For  very  fine  determinations,  a  correction  will  have  to  be 
applied  to  /  owing  to  the  water  being  acidulated  ;  but  for 
ordinary  purposes,  the  tension  may  be  taken  to  be  the  same 
as  that  exerted  by  the  vapour  of  pure  water. 

If  it  is  desired  to  produce  the  decomposition  rapidly,  the 
electrolyte  should  be  so  prepared  as  to  be  of  maximum  con- 
ductivity, its  density  being  then  about  1.25  at  15°  C.  It  is 
not  necessary  that  the  liquid  should  stand  at  the  same  level 
both  inside  and  outside  the  tube  E  when  the  circuit  is  broken. 
By  adopting  this  procedure,  however,  the  necessity  of  finding 
the  density  of  the  electrolyte  after  each  measurement  is 
removed. 


COPPER   VOLTAMETER. 


2I9 


An  improved  form  of  hydrogen  voltameter  is  shewn  in 
Fig.  24.  The  graduated  tube 
E  narrows  down  at  its  upper 
extremity  until  the  opening  is 
of  capillary  dimensions.  It 
then  expands  into  a  little  bulb, 
and  afterwards  bends  over  so 
as  to  rest  on  a  support.  It 
is  filled  by  aspiration  through 
the  rubber  tube  T,  and  the 
capillary  opening  possesses  the 
property  of  permitting  this, 
and  yet  at  the  same  time  pre- 
venting the  escape  of  the  gas, 
provided  that  a  little  of  the 
liquid  is  left  in  the  bulb.  A 
glass  collar  M  attached  to  the 
vase  G,  when  filled  with  water, 
keeps  the  gas  in  the  tube  at  / 
a  uniform  temperature,  and 
enables  it  to  be  determined 
exactly.  Fig.  24. 

IX.    COPPER  VOLTAMETER. 

In  finding  the  intensity  of  an  electric  current  by  means  of 
the  hydrogen  voltameter,  errors  of  temperature,  pressure,  and 
volume  are  almost  certain  to  be  met  with,  and  for  this  reason, 
the  results  obtained  are  often  not  as  accurate  as  is  desirable. 
When,  however,  current  strength  is  measured  by  the  amount 
of  a  metal  deposited,  the  determinations  are  more  accurate, 
since  the  method  is  one  of  direct  weighing.  It  is,  besides, 
more  convenient  and  more  cleanly  in  operation,  and  is  therefore 
generally  ado'pted.  The  metals  commonly  deposited  are  copper 
and  silver,  and  the  solutions  employed  are  copper  sulphate  and 
nitrate  or  chlorate  of  silver. 


220 


EXPERIMENTAL   PHYSICS. 


When  a  copper  voltameter  is  used,  care  must  be  taken  to 
have  it  of  such  form  that  the  density  of  the  current,  i.e.  the 
quotient  of  the  intensity  by  the  surface  of  the  electrodes,  is 
very  small.  Otherwise,  the  deposit  is  made  in  little  globules, 
which  either  do  not  adhere  at  all,  or  if  they  do  they  fall  off  on 
the  slightest  disturbance. 

One  of  the  best  forms  of  the  instrument  is  exhibited  in  Fig. 
25,  where  the  copper  plates  A  and  B,  which  are  partly  immersed 
in  a  solution  of  copper  sulphate,  are  the  positive  and  negative 
electrodes,  respectively.     The  solution  should  be  in  the  propor- 
tion of  five  parts  of  water  to  one 
of   copper    sulphate,    and    should 
contain  a  slight  trace  of  sulphuric 
acid. 

In  setting  up  the  voltameter, 
both  electrodes  must  be  thor- 
oughly cleaned  by  first  rubbing 
them  with  fine  emery  paper,  or 
dipping  them  in  a  solution  of 
nitric  acid,  and  afterwards  rinsing 
them  for  some  time  in  running 
water. 

The  curved  plate  A  is  then 
placed  in  position  as  indicated  in 
the  figure.  B  is  well  dried  by 
rolling  it  on  blotting  paper  and  then  placing  it  in  a  hot-air  bath. 
In  case  the  time  in  which  the  experiment  may  be  performed 
is  limited,  the  plate  may  be  rapidly  dried  by  dipping  it  in 
strong  alcohol  before  rolling  it  in  the  blotting  paper.  Any 
moisture  which  may  adhere  to  it  after  this  will  evaporate  very 
rapidly. 

After  weighing  it  accurately  to  the  tenth  of  a  milligram,  this 
plate  is  attached  to  its  support  and  lowered  into  the  solution 
until  each  point  on  its  surface  is  at  about  the  same  distance 
from  some  portion  of  the  positive  electrode. 


Fig.  25. 


COPPER   VOLTAMETER.  221 

The  circuit  should  also,  in  this  case,  contain  a  mercurial  con- 
tact breaker,  and  the  connections  be  so  made  that  the  current 
will  enter  the  voltameter  by  A  and  leave  by  B.  In  making 
a  test,  the  current  should  be  allowed  to  pass  for  at  least  fifteen 
minutes,  and  especial  care  should  be  taken  to  note  this  time 
accurately. 

On  removing  the  plate  B  from  the  voltameter  after  the  de- 
posit has  been  made,  the  various  operations  already  given  in 
regard  to  thoroughly  cleaning  and  drying  it  are  to  be  repeated. 
Care  must  also  be  taken  while  doing  this  to  handle  the  plate 
very  carefully,  else  portions  of  the  copper  may  be  detached. 

The  amount  of  metal  deposited  is  found  by  again  weighing 
the  plate.  If  W  is  this  amount  in  grams,  the  average  intensity 
of  the  current  passing  during  the  test  is  given  in  amperes  by 

TJT 

C= »  where  /  is  the  number  of  seconds  the  circuit  was 

tx.  000328 

complete,  and  .000328  grams  is  the  electro-chemical  equivalent 
of  copper. 

Theoretically,  it  should  make  no  difference  in  which  direction 
the  current  is  passed  through  the  voltameter,  as  the  plate  B  in 
the  one  case  should  lose  the  same  amount  that  it  gains  in  the 
other.  It  is  found,  however,  that  secondary  chemical  actions 
take  place  in  the  neighbourhood  of  the  positive  plate,  which 
cause  a  loss  in  it  without  a  corresponding  deposit  on  the  nega- 
tive electrode.  '  For  this  reason  the  amount  of  metal  deposited 
rather  than  that  dissolved  is  taken  as  a  measure  of  the  current 
passing. 

A  modification  of  the  voltameter  just  described  is  obtained 
by  using  spiral  coils  instead  of  plates  for  electrodes.  The  two 
coils  being  of  different  sizes  are  placed  one  within  the  other  in 
the  solution,  the  negative  generally  being  placed  in  the  center. 
This  form  of  the  instrument  possesses  the  advantage  of  present- 
ing a  very  large  surface  to  the  action  of  the  liquid. 

In  finding  the  strength  of  a  current  by  the  amount  of  silver 
deposited,  Poggendorff's  voltameter  is  generally  used.  The  nega- 


222  EXPERIMENTAL   PHYSICS. 

tive  electrode  is  a  platinum  bowl  resting  on  a  metallic  plate  to 
which  the  leading  wire  is  attached. 

The  solution  is  placed  in  this  vessel,  and  the  positive  elec- 
trode, which  is  a  disc  of  silver,  is  suspended  in  it  so  that  its 
edge  is  equidistant  from  the  sides  and  the  bottom  of  the  bowl. 

It  is  customary  to  surround  the  silver  disc  by  a  small  bag 
made  of  fine  gauze  or  filter  paper,  so  that  the  particles  of  metal 
which  are  separated  by  other  than  electrolytic  action  may  be 
prevented  from  being  deposited  on  the  platinum. 

The  solution  used,  in  this  voltameter  contains  from  fifteen  to 
twenty  per  cent  of  silver  nitrate. 


X.  CALIBRATION  OF  GALVANOMETERS. 

In  Experiment  VII.  it  has  been  shewn  that  when  an  electric 
current  is  measured  by  passing  it  through  a  galvanometer  whose 
dimensions  can  be  readily  determined,  its  intensity  is  given 
by  a  relation  either  of  the  form  C=Ktan0,  or  of  C—K  sin#, 
according  to  the  disposition  adopted. 

Many  galvanometers,  however,  are  constructed  in  such  a 
manner  that  although  their  deflections  follow  the  tangent,  or 
the  sine  law,  yet  it  is  difficult  and  inconvenient  to  ascertain 
the  number  of  windings  in  the  coil,  and  quite  impossible  to 
obtain  even  an  approximation  to  its  mean  radius. 

The  constant  K,  which  depends  on  these  quantities  and  on 
the  intensity  of  the  terrestrial  field,  cannot  then  be  calculated, 
and  must  be  determined  experimentally. 

METHOD  I.  —  A  standard  tangent  galvanometer  whose  con- 
stant KI  is  known  is  joined  in  series  in  a  circuit  with  the  one 
examined. 

If  for  a  given  current  the  deflections  of  the  two  instruments 
are  respectively  0l  and  6,  then  we  have 


CALIBRATION    OF    GALVANOMETERS. 


223 


as  a  relation  for  determining  the  constant.  The  value  of  K 
should  be  found  by  taking  the  mean  of  a  number  of  readings 
obtained  by  varying  the  resistance  in  the  circuit,  and  by  revers- 
ing the  direction  of  the  current  through  the  galvanometers. 
When  this  method  is  followed,  care  must  be  taken  to  place  the 
instruments  at  a  considerable  distance  apart,  else  the  needle 
of  the  one  may  be  affected  by  the  magnetic  field  of  the  other. 

METHOD  II.  —  In  this  method  the  standard  galvanometer  of 
the  last  is  replaced  by  a  copper  or  a  silver  voltameter  of  one  of 
the  forms  described  in  Experiment  IX.  The  current  passed 
through  is  kept  as  steady  as  possible  by  means  of  a  variable 
resistance,  and  any  variations  which  do  occur  are  corrected  by 
taking  the  mean  of  the  deflections  observed  at  stated  intervals 
during  the  test. 

If    6   is    the    mean    deflection,    the    constant    is    given    by 

K= ,  where  M  is  the  amount  of  the  metal  deposited  in 

zt  tan  6 

grams,  z  its  electro-chemical  equivalent,  and  /  the  number  of 
seconds  the  current  was  passing.  For  very  exact  determina- 
tions the  test  should  be  continued  for  at  least  two  hours,  and 
the  circuit  so  arranged  that  the  deflection  on  the  galvanometer 
is  about  45°,  this  being  the  deflection  which  gives  the  most 
accurate  results. 

METHOD  III.  —  This,  which  is  a  very  simple  method,  and 
probably  the  most  rapid,  is  a  direct  application  of  Ohm's  law. 
A  battery  or  cell  of  constant  electromotive  force  is  joined  up 
in  circuit  with  the  galvanometer,  and  the  current  is  varied  by 
the  insertion  of  one  or  more  of  a  set  of  graduated  resistances. 
If  the  electromotive  force  of  the  cell  is  denoted  by  E,  and  the 
deflection  of  the  galvanometer  corresponding  to  a  given  current 

intensity  is  0,  the  constant  is  given  by  K=  — -,  where  R  is 

J\.  tan  6 

the  total  resistance  in  the  circuit.  Usually  the  inserted  resist- 
ance is  very  high,  and  in  that  case  the  resistances  of  the  bat- 
tery and  the  galvanometer  may  be  neglected ;  otherwise  these 
must  be  determined  by  some  one  of  the  methods  given  later. 


224 


EXPERIMENTAL   PHYSICS. 


It  frequently  happens  that  the  current  cannot  be  readily 
reduced  to  the  proper  intensity  by  the  insertion  of  resistances 
directly  in  the  circuit.  The  galvanometer  should  then  be  pro- 
vided with  a  variable  shunt,  since  by  suitably  adjusting  it  any 
desired  modification  of  the  intensity  can  be  easily  obtained. 
The  current  in  the  main  circuit  being  known,  that  passing 

through  the  galvanometer  can 
then  be  calculated  by  the  law  of 
divided  circuits. 

In  many  experiments  the  in- 
vestigation does  not  consist  in 
ascertaining  the  intensity,  but 
rather  in  detecting  the  presence 
of  weak  currents,  and  in  deter- 
mining their  direction. 

Galvanometers  which  are  used 
for  this  purpose  are  of  special 
construction,  and  are  so  designed 
as  to  be  extremely  sensitive. 

One  of  the  most  useful  of  this 
class  is  that  known  as  Thomson's 
(Fig.  26). 

The  magnets  which  form  an 
astatic  couple  are  very  small,  and 
are  connected  by  a  fine  strip  of 

aluminium,  which  is  itself  suspended  from  the  support  V  by  a 
strand  of  unspun  silk.  This  strip  carries  a  thin  plate  of  mica  for 
damping  the  oscillations  of  the  magnets,  and  a  small  concave 
mirror  attached  to  it  serves  to  indicate  the  sense  of  the  deflections. 
The  coils,  which  are  so  arranged  that  a  magnet  is  at  the 
center  of  each,  are  divided  into  two  halves,  and  these  are  sup- 
ported on  two  brass  plates,  one  of  which,  LL',  is  shewn  in  the 
figure.  The  binding  poles  BB'  serve  to  connect  the  two  parts 
of  each  coil. 

A  large  steel  bar  feebly  magnetized  is  supported  above  the 


Fig.  26. 


CALIBRATION  OF  GALVANOMETERS. 


instrument,  and  as  it  can  rotate  about  the  vertical,  or  slide  up 
and  down,  the  earth's  magnetic  field  and  the  sensibility  of  the 
instrument  can  be  modified  as  desired. 

On  account  of  the  great  sensitiveness  of  this  instrument  it 
is   generally   used   in    experiments  of   precision.      It    however 
requires  very  delicate  han- 
dling, and  must  be  far  re- 
moved from  the  presence 
of  moving  masses  of  iron. 

Another  form  of  galva- 
nometer which  is  very  sen- 
sitive, but  which  is  more 
stable,  is  that  of  Depretz 
and  D'Arsonval.  It  is 
shewn  in  Fig.  27. 

It  is  the  coil  which 
moves  in  this  instrument 
and  not  the  magnet.  The 
former  is  made  up  of  a 
number  of  turns  of  very 
fine  wire,  carries  a  small 

concave  mirror,  and  is  suspended  by  two  silver  wires  between 
the  poles  of  a  horseshoe  magnet,  as  shewn  in  the  figure.  These 
wires  also  serve  to  lead  the  current  to  and  from  the  coil.  A 
tube  of  soft  iron,  which  is  held  within  the  coil,  and  moves  with 
it,  serves  to  increase  the  intensity  of  the  magnetic  field. 

Owing  to  induction  the  damping  in  this  instrument  is  so 
rapid  that  when  a  current  is  passed  through  the  coil  it  almost 
immediately  assumes  its  position  of  rest. 

When  observations  are  made  with  these  galvanometers  two 
different  arrangements  can  be  made.  For  very  exact  work,  that 
shewn  in  Fig.  13  is  adopted.  There  a  scale  is  placed  in  front 
of  the  mirror,  and  the  images  of  its  graduations  are  observed 
by  means  of  a  telescope  whose  axis  lies  in  the  perpendicular 
from  the  center  of  the  mirror  on  the  scale. 


Fig.  27. 


226 


EXPERIMENTAL   PHYSICS. 


Ordinarily,  when  rapid  determinations  are  necessary  the  scale 
is  ruled  on  some  translucent  substance  such  as  celluloid,  and  is 
supported  on  a  screen.  In  this  screen  there  is  a  small  opening 
immediately  below  the  scale,  with  a  dark  thread  stretched  verti- 
cally across  it.  When  this  opening  is  illuminated  with  parallel 
rays  of  light  from  some  source  an  image  is  formed  by  the 
mirror  on  the  scale,  and  any  displacement  of  this  image  will  at 
once  indicate  the  deflections  of  the  instrument.  In  order  to 
have  a  distinct  image  when  the  latter  method  is  adopted,  it  is 
best  to  place  the  scale  at  a  distance  from  the  mirror  equal  to 
the  length  of  its  radius  of  curvature. 


Fig.  28. 

Galvanometers  of  this  class  may  also  be  calibrated  to  indi- 
cate absolute  intensity.  Different  currents  are  passed  through 
the  instrument,  and  their  strengths  are  determined  by  one  of  the 
methods  given  above.  Curves  are  then  plotted  by  taking  the 
deflections  as  indicated  by  the  scale  graduations  for  abscissae, 
and  the  corresponding  currents  for  ordinates.  By  a  reference 
to  these  curves,  the  intensity  of  a  current  can  readily  be  deter- 
mined when  the  deflection  it  produces  is  known. 

In  Fig.  28  there  is  shewn  one  of  a  class  of  direct  reading 
galvanometers,  usually  called  ammeters,  since  they  are  graduated 
to  give  the  value  of  the  current  strength  in  amperes.  The 
principle  of  this  instrument  can  be  easily  understood  by  a  refer- 
ence to  Fig.  29. 


GALVANIC   BATTERIES. 


227 


A  powerful  magnetic  field  is  formed  by  two  semicircular  per- 
manent magnets,  whose  like  poles  are  A  A'  and  BB\     A  double 
coil  of  wire  CO  is   fixed   obliquely  between  the  two   pairs  of 
poles,   and   at   its    center   there   is 
pivoted  a  small    bar   of    soft   iron, 
which  carries  the  pointer  indicated 
in  Fig.  28.     This  bar  always  takes 
up   a   position    with    its   axis   coin- 
ciding  with    the    direction    of   the 
lines  of  force,  and  when  a  current 
is  passing  through  the  coil,  the  mag- 
netic field  which  it  produces  modi- 
fies that  of  the   magnets,  and  the 
bar  is  therefore  deflected. 

In  these  instruments  the  inten- 
sities corresponding  to  different 
deflections  are  recorded  on  the  dial ;  and  as  the  accuracy 
of  the  graduations  depends  on  the  magnets  retaining  their 
initial  strength,  they  should  be  tested  from  time  to  time,  and, 
if  necessary,  recalibrated.  The  range  of  the  instrument  is 
extended  by  attaching  to  it,  by  the  brass  strips  m  and  n,  a  shunt 
CD  (Fig.  28),  whose  resistance  bears  a  simple  ratio  to  that  of 
the  instrument  proper. 


Fig.  29. 


XI.    GALVANIC   BATTERIES. 

The  following  article  is  devoted  to  short  descriptions  of  a 
few  of  the  cells  commonly  used  in  electrical  laboratories.  No 
attempt  is  made  to  trace  the  development  of  this  part  of  the 
subject,  and  complete  details  are  omitted,  as  these  can  be  found 
in  any  one  of  the  many  treatises  dealing  at  length  with  this 
department  of  electrical  investigation. 

A  few  exercises  are  appended,  however,  for  the  purpose  of 
illustrating  fundamental  notions  in  regard  to  the  action  of  cells 
and  the  flow  of  currents.  Table  XX.  contains  the  electromotive 


228 


EXPERIMENTAL   PHYSICS. 


forces  of  a  number  of  elements,  and  Table  XIX.,  their  resistances. 
As  the  latter  depend  very  much  on  the  dimensions  of  the  cell 
and  on  the  strengths  of  the  solutions  used,  the  results  are, 
of  course,  only  approximate. 

When  possible,  the  laboratory  should  be  provided  with  a 
dynamo,  as  currents  generated  in  this  way,  especially  if  of 
strong  intensity,  are  produced  at  a  much  smaller  cost  than  by 
the  use  of  batteries. 

I.  Daniell's  Cell.  Meidinger  type.  —  In  this  cell  there  is  an 
outer  glass  vessel  made  of  two  cylinders  connected  by  a  shoul- 
der, a  glass  tumbler  resting  on 
the  bottom  of  this  vessel,  and  an 
inverted  flask  whose  volume  is 
about  two  liters. 

In  setting  up  the  cell,  the  posi- 
tive pole,  consisting  of  a  copper 
coil  or  a  thin  plate  of  that  metal 
rolled  into  a  cylinder,  is  placed  in 
the  tumbler  and  then  covered  with 
a  saturated  solution  of  copper  sul- 
phate. The  flask,  after  being  filled 
as  far  as  possible  with  crystals  of 
this  substance,  and  the  balance  of 
the  space  with  the  saturated  solu- 
tion, is  inverted,  and  placed  with 
its  mouth  under  the  surface  of  the  liquid  in  the  tumbler.  The 
negative  pole,  a  cylinder  of  zinc,  is  next  placed  in  position  as 
shewn  in  the  figure,  and  after  both  tumbler  and  flask  are  lowered 
into  the  outer  vessel  a  ten  per  cent  solution  of  zinc  sulphate  is 
gently  poured  into  the  latter  by  means  of  a  rubber  tube  until  it 
covers  the  zinc  plate.  Owing  to  the  density  of  this  liquid  being 
less  than  that  of  the  copper  sulphate,  it  will,  when  not  agitated, 
float  on  the  top  of  the  latter  in  the  tumbler  without  mixing. 
Wires  to  which  clamps  may  be  attached  lead  from  the  two 


Fig.  30. 


GALVANIC    BATTERIES. 


229 


electrodes,  that  from  the  positive  being  insulated  by  gutta- 
percha  in  order  to  prevent  its  being  eaten  away  by  chemical 
action  at  the  surface  of  separation  of  the  liquids. 

In  this  cell,  the  sulphuric  acid  acting  on  the  negative  elec- 
trode forms  zinc  sulphate  and  liberates  hydrogen.  The  latter 
unites  with  the  copper  sulphate  to  form  sulphuric  acid,  which 
is  again  ready  to  react  upon  more  zinc,  while  the  copper,  which 
is  set  free,  is  deposited  on  the  positive  plate. 

The  reactions  are  expressed  in  the  following  equations : 

Zn  +  H2SO4  =  ZnSO4  +  2  H, 
2H  +  CuSO4=Cu  +  H2SO4. 

In  mounting  this  cell,  and,  in  fact,  all  cells,  care  should  be 
taken  to  have  all  the  chemicals  pure,  and  to  use  only  distilled 
water  in  making  the  solutions.  Ordinary  commercial  zinc  con- 
tains many  impurities,  such  as  iron  and  lead,  so  that  when  a 
plate  of  such  metal  is  used  in  a  cell  these  particles  form  with 
the  adjacent  zinc  a  series  of  small  voltaic  elements,  and  local 
actions  are  set  up  which  waste  away  the  zinc  without  doing  any 
useful  work.  This  waste  may  be  avoided,  and  the  plate  made 
to  act  practically  as  if  it  were  pure  by  covering  it  with  an 
amalgam.  This  is  done  by  plunging  it  for  a  few  seconds  in 
dilute  sulphuric  acid,  and  then  rubbing  it  over  with  mercury. 
After  doing  this  it  should  be  thoroughly  washed  before  being 
used.  Instead  of  having  the  positive  plate  of  this  cell  all  of 
copper,  it  is  sometimes  replaced  by  one  of  lead  which  has  been 
copper-plated.  This  modification  possesses  the  advantage*  of 
readily  permitting  the  removal  of  the  copper  deposit.  With 
ordinary  usage  a  cell  set  up  as  described  will  last  without 
renewal  for  about  a  year,  and  as  it  has  a  very  constant  electro- 
motive force,  and  produces  a  very  steady  current,  it  is  admira- 
bly adapted  for  laboratory  work. 

II.  Grove  Cell. — This  element  in  its  common  form  consists 
of  a  rectangular  outer  vessel  AE,  made  of  porcelain,  or  ebonite, 


230 


EXPERIMENTAL   PHYSICS. 


;i  U-shaped  plate  B  of  amalgamated  zinc,  a  porous  cup  C,  and  a 

thin  strip  of  platinum  D. 

Sulphuric  acid  diluted  in  the  ratio  of  one  to  ten  is  placed  in 

the  outer  vessel,  and  concentrated  nitric  acid  in  the  porous  cup. 
The  platinum  and  the  zinc  form  the  posi- 
tive and  the  negative  poles  of  the  cell 
respectively.  The  following  equations  ex- 
press the  reactions  for  the  element : 

4  +  2H, 


When  the  cell  has  been  allowed  to  act 
for  some  time  the  nitric  acid  becomes 
used  up  by  the  chemical  action,  and  the 
sulphuric  acid  mixed  with  zinc  sulphate. 

There  is  a  consequent  increase  in  the  internal  resistance,  but 
since  this  cell  has  a  high  electromotive  force,  it  is  capable  of 
producing  a  strong  current  for  a  considerable  time. 

The  U  form  of  the  zinc  is  adopted  in  order  to  thereby  reduce 
the  distance  between  the  electrodes,  but  it  is  not  very  economical 
as  the  zinc  is  almost  invariably  eaten  away  at  its  lowest  point 
first,  and  a  large  quantity  of  the  metal  is  thereby  rendered 
useless. 

The  chief  objections  to  an  extensive  use  of  Grove  cells  arise 
from  the  high  price  of  platinum,  the  injurious  effects  produced 
by  the  nitric  peroxide  gas  given  off,  and  from  the  nitric  acid 
diffusing  through  the  porous  cup  and  acting  directly  on  the 
zinc.  The  element  is,  however,  an  extremely  convenient  one, 
and  is  generally  used  when  a  strong  current  is  required. 

III.  Grenet  Cell. — This  cell  as  ordinarily  constructed  con- 
sists of  a  large  glass  bottle,  whose  volume  is  about  two  liters, 
and  whose  shape  is  similar  to  that  shewn  in  Fig.  32.  A  cover 

*  The  oxidizing  action  may  go  much  beyond  this  phase,  and  can  be  represented 
by  the  equation  2  H%  HNO3  =  H2O  +  HNO,. 


GALVANIC    BATTERIES. 


231 


of  ebonite  rests  on  the  brass  collar  surrounding  the  upper 
portion  of  the  neck,  and  carries  two  parallel  slabs  C  and  C  of 
carbon  which  form  the  positive  plates  of  the  cell.  The  negative 
plate  Z  is  made  of  amalgamated  zinc,  and  is  attached  to  a  brass 
rod  a  which  can  slide  freely  through  a  socket 
in  the  cover,  and  so  permits  this  electrode  to  be 
raised  out  of  the  liquid  when  the  cell  is  not  in 
use. 

A  piece  of  ebonite  attached  to  the  inside 
of  each  of  the  carbon  plates  serves  to  guide 
the  zinc  plate  when  it  is  being  lowered,  and 
at  the  same  time  prevents  contact. 

The  solution  for  the  cell  is  composed  of 
bichromate  of  potash,  sulphuric  acid,  and  water. 
Various  proportions  have  been  suggested,  but 
the  following  composition  is  considered  one  of  Fig  32. 

the  best : 


Water, 


80  parts  by  weight. 


Pulverized  bichromate  of  potash,  12          „         „ 
Sulphuric  acid,  36          „         „ 

On  mixing  these  together  chromic  acid  and  potassium  sul- 
phate are  formed,  and  when  the  zinc  is  lowered  and  the  current 
established  the  chemical  action  consists  in  the  chromic  acid 
combining  with  the  sulphuric  to  form  chromium  sulphate,  and 
the  oxygen,  which  is  then  set  free,  uniting  with  the  hydrogen 
to  form  water. 

The  reactions  are  as  follows  : 

H2O  +  H2S04  +  K2Cr2O7  =  K2SO4  +  2  H2CrO4, 
3  Zn  +  3  H2S04=3  ZnSO4  +  6  H, 

6H  +  3  H2S04  +  2  H2Cr04  =  Cr2(SO4)3  +  8  H2O. 

When  fresh  the  solution  is  of  an  orange  colour,  but  as  the 
potash  bichromate  becomes  used  up  it  changes  to  a  green 


232 


EXPERIMENTAL    PHYSICS. 


or  blue  owing  to  the  formation  of  chromic  sulphate  which 
combines  with  the  potassium  sulphate  to  form  chrome  alum.* 
As  a  consequence  the  current  becomes  rapidly  weaker  and 
more  bichromate  should  then  be  added.  If,  however,  the 
action  becomes  weak  without  this  change  of  colour,  either  the 
supply  of  sulphuric  acid  has  become  exhausted,  and  should 
be  renewed,  or  the  carbons  have  become  coated  with  hydrogen 
bubbles.  These  may  be  removed  either  by  the  chemical  action 
of  the  solution  if  the  circuit  be  broken  for  a  short  time,  or 
by  violently  shaking  up  the  liquid. 

This  element  when  it  is  first  set  up  has  a  high  electromotive 
force.  It  is  well  suited  for  the  excitation  of  induction  coils,  but 
polarizes  very  rapidly.  In  some  forms  of  the  bichromate  ele- 
ment the  zinc  plate  is  not  movable,  and  local  action  is  pre- 
vented by  separating  it  from  the  chromic  acid  by  means  of  a 
porous  cup. 

IV.  Leclancht  Cell.  —  The  Leclanche"  cell  in  its  original  form 
is  represented  in  Fig.  33,  where  A  is  a  square  glass  vessel,  B, 
the  negative  electrode,  is  a  rod  of  amalgamated  zinc,  C,  a  porous 


Fig.  33. 


Fig.  34. 


cup,  contains  equal  quantities  of  peroxide  of  manganese  and 
small  pieces  of  carbon,  and  D,  the  positive  plate,  is  a  strip  of 
carbon  inserted  in  the  mixture. 

The  exciting  liquid  is  a  saturated  solution  of  sal  ammoniac, 
*KCr(SO4)2+  I2H2O. 


GALVANIC    BATTERIES. 


233 


which  about  half  fills  the  vessel  A.  When  the  current  is  pass- 
ing it  acts  on  the  zinc,  and  forms  hydrogen,  ammonia,  and  zinc 
chloride.  The  peroxide  of  manganese  is  slowly  reduced  to  a 
sesquioxide,  and  the  oxygen  given  off  unites  with  the  hydrogen. 
The  following  are  the  reactions  : 

2  HN4Cl  +  Zn  =  ZnCl2  +  2  NH3  +  2  H,  (i) 

2H  +  2MnO2=Mn2O3-t-H2O.  (2) 

The  current  from  this  cell  falls  off  very  rapidly  on  closing  the 
circuit,  but  as  the  element  has  the  property  of  rapidly  building 
itself  up,  it  is  exceedingly  well  adapted  for  open  circuit  work, 
such  as  the  ringing  of  electric  bells. 

The  strip  of  carbon  is  generally  surmounted  by  a  lead  cap,  to 
which  the  leading  wire  may  be  attached,  and  in  order  to  prevent 
the  liquid  rising  by  capillary  action,  and  increasing  the  resist- 
ance of  the  cell  by  its  action  on  the  lead,  the  upper  part  of  the 
strip  is  soaked  in  melted  paraffine.  This  on  cooling  solidifies  in 
the  pores,  but  on  its  being  carefully  removed  from  the  outside 
of  the  carbon,  a  good  conducting  surface  is  obtained,  and,  at  the 
same  time,  capillary  action  is  prevented. 

This  cell  possesses  many  important  advantages  over  other 
elements.  It  has  a  high  electromotive  force  and  a  compara- 
tively low  resistance ;  the  materials  used  in  its  construction  can 
be  obtained  at  a  very  low  price ;  it  emits  no  noxious  gases  and 
contains  no  poisonous  liquids.  The  zinc  is  acted  on  by  the  sal 
ammoniac  only  when  the  current  is  passing,  and,  as  the  liquid 
freezes  at  a  very  low  temperature,  the  element  is  much  used  in 
very  cold  climates. 

The  cell  as  constructed  now  is  shewn  in  Fig.  34.  There  the 
porous  cup  is  dispensed  with,  and  the  carbon  electrode  is  held 
between  two  plates  made  of  carbon,  manganese  dioxide,  and 
shellac,  by  subjecting  the  mixture  to  very  high  pressures.  The 
zinc  is  also  placed  close  to  these  plates,  and  the  whole  is  held 
together  by  rubber  bands. 


234 


EXPERIMENTAL   PHYSICS. 


This  modification  has  a  somewhat  lower  internal  resistance, 
and  a  slightly  higher  electromotive  force  than  the  cell  as 
originally  devised. 

V.  Latimer-Clark 's  Cell.  —  Many  efforts  have  been  put  forth 
to  devise  some  form  of  cell  which  would  possess  a  constant 
electromotive  force,  and  could  therefore  be  adopted  as  a  stan- 
dard for  comparative  purposes.  That  proposed  by  Latimer- 
Clark  more  nearly  satisfies  the  requirements  than  any  other  as 
yet  constructed.  It  is  prepared  by  dissolving  sulphate  of  zinc 
in  boiling  distilled  water.  After  cooling,  this  is  decanted  and 
the  saturated  solution  thus  obtained  is  used  to  make  a  thick 
paste  with  mercurous  sulphate.  This  is  heated  to  100°  C.  in 
order  to  expel  all  the  air,  and  is  then  poured  upon  the  surface  of 
warm  mercury  contained  in  some  vessel  of  approved  form.  A 
zinc  plate  is  then  suspended  in  this  paste,  and  the  whole  sealed 
in  by  pouring  over  it  melted  paraffine.  The  mercury  and  the 
zinc  are  the  positive  and  the  negative  electrodes  respectively, 
and  each  has  a  conducting  wire  leading  from  it,  that  from  the 
mercury  being  of  platinum,  and  passing  up  through  a  zinc  glass 
tube  embedded  in.  the  mixture.  The  reaction  for  the  Clark 
cell  is  given  by  the  relation  2  H  +  Hg2SO4  =  2  Hg  +  H2SO4. 

The  electromotive  forces  of  cells  prepared  in  this  way  are 
found  to  be  exceedingly  uniform,  their  variations  not  exceeding 
.0005  volts.  Owing  to  the  possession  of  this  property  they  are 
admirably  suited  for  comparative  work,  and  are  greatly  superior 
to  standard  Daniell  cells,  whose  electromotive  forces  are  found 
to  depend  largely  on  the  concentration  of  the  solutions  used. 

Careful  determinations  have  been  recently  made  by  Messrs. 
Glazebrook  and  Skinner  on  this  form  of  element,  their  experi- 
ments involving  the  examination  of  hundreds  of  cells.  The 
results  of  their  work  shew  that  its  electromotive  force  at  15°  C. 
is  1.4342  volts.  In  constructing  cells  of  this  class  now  the  paste 
is  baked,  so  that  on  cooling  it  becomes  very  hard.  This  has  the 
effect  of  increasing  the  internal  resistance  to  about  15,700  ohms. 


GALVANIC   BATTERIES. 


235 


The  cell  can  then  be  used  in  a  closed  circuit  without  any  appre- 
ciable diminution  in  the  electromotive  force. 

EXERCISE  I.  —  Static  charge. 

Insert  a  plate  of  copper  and  one  of  amalgamated  zinc  in  a 
dilute  solution  of  sulphuric  acid,  and  attach  a  wire  to  each  of 
them  ;  connect  to  the  lower  disc  of  a  condensing  electroscope 
the  wire  leading  from  the  zinc,  and  to  the  upper  disc  that  from 
the  copper  plate. 

Then  remove  the  wires,  and  lift  the  upper  disc  by  means 
of  the  insulated  handle.  If  the  electroscope  be  very  delicate, 
the  gold  leaves  will  diverge,  shewing  that  they  are  electrified. 
The  same  phenomenon  can  be  exhibited  by  interchanging  the 
wires  and  repeating  the  operations  described. 

The  experiment,  which  may  be  conducted  much  more  suc- 
cessfully with  a  battery  of  three  or  four  Grove  cells,  or  a 
dynamo,  indicates  that  when  the  circuit  of  a  current  generator 
is  open,  the  two  terminals  are  statically  charged,  and  are  at  a 
difference  of  potential. 

If  the  laboratory  is  provided  with  a  quadrant  electrometer, 
much  better  results  can  be  obtained  by  using  this  instrument 
instead  of  the  electroscope. 

EXERCISE  II. — Polarization. 

Place,  as  in  Exercise  I.,  a  plate  of  amalgamated  zinc  and  one 
of  copper  in  a  vessel  containing  a  dilute  solution  of  sulphuric 
acid,  and  as  soon  as  the  circuit  is  made  observe  the  current 
strength  as  indicated  by  a  sensitive  galvanometer.  It  will  begin 
to  diminish  the  moment  the  circuit  is  closed.  If,  after  the  cell 
has  been  allowed  to  act  for  about  five  minutes  the  circuit  be 
opened  for  another  five,  it  will  be  found  on  again  closing  it  that 
it  has  regained  almost  entirely  its  initial  intensity. 

In  seeking  the  cause  of  this,  one  has  only  to  watch  closely 
the  action  in  the  element.  As  soon  as  the  current  begins  to 
pass,  hydrogen  is  deposited  in  little  bubbles  on  the  copper  elec- 


236  EXPERIMENTAL   PHYSICS. 

trode,  and,  after  some  time,  these  form  a  gaseous  layer  between 
the  sulphuric  acid  and  the  plate.  If  the  solution  be  shaken  up, 
or  the  plates  agitated  separately  or  together  in  the  liquid,  or  the 
surface  of  the  copper  plate  be  rubbed  over  with  a  brush,  it  may 
be  shewn  that  the  diminution  of  the  current  strength  depends 
only  on  the  presence  of  this  gaseous  layer,  and  that  the  dis- 
appearance of  the  hydrogen,  however  brought  about,  from  the 
positive  electrode,  is  always  accompanied  by  an  increase  in 
the  current. 

As  the  strength  of  the  current  depends  only  on  the  electro- 
motive force  of  the  cell  and  on  its  resistance,  the  presence  of 
the  gas  must  affect  one  or  both  of  these.  That  it  does  affect 
the  resistance  is  evident,  since  the  active  surface  of  the  cop- 
per is  at  once  diminished  when  it  is  present.  It  will  be  seen 
later  that  a  diminution  in  the  electromotive  force  also  takes 
place. 

In  order  to  prevent  this  polarization,  as  it  is  called,  cells 
generally  contain  some  liquid  or  substance  which  will  act 
chemically  on  the  hydrogen.  Examples  of  this  are  found  in 
the  use  of  nitric  acid  in  the  Grove,  bichromate  of  potash  in  the 
Grenet,  and  peroxide  of  manganese  in  the  Leclanch6  cell. 

EXERC i SE  III.  —  Resistance. 

In  this  exercise  a  long  rectangular  wooden  box,  a  plate  of 
copper  and  one  of  zinc,  and  a  set  of  wires  of  homogeneous 
structure  and  of  constant  cross-section,  are  required.  A  solu- 
tion of  sulphuric  acid  is  put  in  the  box,  and  the  two  plates  are 
also  inserted  in  it  at  a  known  distance  apart.  Two  of  the  wires 
are  then  selected,  and  these,  together  with  a  sensitive  galva- 
nometer, whose  coil  is  also  made  of  this  wire,  are  used  to  com- 
plete the  circuit.  The  initial  intensity  of  the  current  produced 
is  noted  and  then  modifications  of  this  arrangement  are  made  by 
altering  the  distance  between  the  plates,  or  the  amount  of  their 
surface  immersed,  and  also  by  using  different  lengths  of  wires 
and  wires  of  different  diameters. 


GALVANIC    BATTERIES. 


237 


If  the  initial  intensity  of  the  current  corresponding  to  each 
disposition  be  carefully  noted,  it  will  be  found  that  the  results 
obtained  go  to  establish  the  law  that  the  current  produced  by  a 
cell  varies  inversely  as  the  resistance  of  the  circuit  including 
that  of  the  cell,  and  that  the  resistance  of  a  uniform  conductor 
of  any  substance  varies  directly  as  its  length  and  inversely  as 
its  cross-section. 

EXERCISE  IV.  —  Electromotive  Force. 

Many  experiments  establish  the  fact  that  when  two  electri- 
fied bodies  which  are  at  a  difference  of  potential  are  brought 
close  together,  or  are  connected  by  a  wire  or  other  conductor, 
a  discharge  or  current  passes  between  them,  and  they  are  in 
consequence  reduced  to  the  same  electrical  state. 

In  Exercise  I.,  it  has  been  shewn  that  a  difference  of  poten- 
tial always  exists  between  the  terminals  of  a  battery  or  cell  in 
open  circuit.  In  this  case,  however,  the  current  produced  when 
the  terminals  are  joined  does  not  consist  of  a  single  discharge, 
and  is  not  followed  by  the  establishment  of  electrical  equilib- 
rium between  the  two  electrodes.  The  current  is  continuous 
and  lasts  as  long  as  chemical  action  is  kept  up  in  the  cell ; 
while  the  difference  of  potential  which  this  action  produces 
between  the  terminals  of  the  battery  when  the  current  is  estab- 
lished is,  owing  to  the  existence  of  this  current,  intermediary 
between  the  potential  difference  in  open  circuit  and  the  zero 
difference  which  obtains  when  the  element  ceases  to  act. 

Owing  to  a  battery  or  cell  possessing  this  property  of  main- 
taining its  electrodes  at  a  difference  of  potential  when  they  are 
connected  by  a  conductor,  it  is  said  to  be  the  seat  of  an  electro- 
motive force. 

The  electromotive  force  of  an  element  which  may  thus  be 
considered  to  be  the  cause  of  the  current  is  measured  by  the 
greatest  difference  of  potential  which  exists  between  its  termi- 
nals when  the  circuit  is  open  or  when  the  circuit  contains  a 
resistance  so  great  that  practically  no  current  passes. 


238  EXPERIMENTAL   PHYSICS. 

It  is  independent  of  the  dimensions  of  the  cell,  and  depends 
only  on  the  materials  which  enter  into  its  composition.  In 
order  to  make  clear  that  this  is  so,  it  will  suffice  to  perform  a 
few  experiments  with  cells  of  different  types  and  of  different 
dimensions. 

If  two  zinc-copper  elements,  of  the  type  described  in  Exercise 
I.,  be  constructed  exactly  similar  in  every  way,  the  currents 
produced  by  them  will  be  practically  of  the  same  intensity.  If, 
then,  they  be  joined  in  opposition,  i.e.  so  that  they  tend  to  send 
currents  in  opposite  directions  in  the  same  circuit,  it  will  be 
found  that  no  current  is  produced.  If,  now,  instead  of  having 
the  two  cells  of  exactly  the  same  dimensions,  the  plates  of  one 
of  them  are  made  much  larger  than  those  of  the  other,  the  larger 
cell  will  (the  external  circuit  being  the  same)  produce  much  the 
stronger  current.  When  the  two  cells  are  opposed  to  each 
other,  however,  there  will  again  be  no  current  produced,  owing 
to  the  fact  that,  although  the  cells  are  of  different  dimensions, 
their  electromotive  forces  are  the  same. 

Further,  if,  instead  of  performing  this  experiment  with  two 
zinc-copper  elements,  one  of  these  be  used  and  an  iron-copper 
element  of  exactly  the  same  dimensions  be  taken  for  the  other, 
it  will  be  found  that  when  these  are  joined  in  opposition  there 
is  a  current  produced  which  takes  the  same  direction  as  it  would 
if  the  zinc-copper  element  only  were  acting. 

From  this  it  follows  that  the  electromotive  force  of  the  latter 
is  greater  than  that  of  the  iron-copper  element,  and  further  that 
the  materials  of  which  a  cell  is  composed  determines  its  electro- 
motive force.  In  order  to  make  this  still  clearer,  the  last  experi- 
ment can  be  modified  by  using  a  small  zinc-copper  element  and 
a  very  large  iron-copper  one.  On  joining  these  in  circuit  in 
opposition,  a  current  is  again  obtained  whose  direction  is  the 
same  as  that  of  the  current  obtained  in  the  preceding  experi- 
ment. Many  similar  exercises  can  be  readily  devised  by  using 
different  exciting  liquids  and  plates  of  various  other  materials. 
These,  however,  will  be  sufficient  to  substantiate  the  statement 


DETERMINATION    OF   RESISTANCE. 


239 


given  above,  that  it  is  the  materials  used  in  t/ie  construction  of 
an  element,  and  not  its  dimensions,  which  determine  its  electro- 
motive force. 

The  method  adopted  in  these  experiments  can  be  applied 
with  advantage  to  shew  that  when  polarization  takes  place  in 
an  element,  there  is  a  consequent  diminution  in  its  electro- 
motive force  as  well  as  an  increase  in  its  resistance.  To  do  this, 
it  is  only  necessary  to  oppose  a  freshly  prepared  cell  to  one 
exactly  similar  in  dimensions  and  composition  which  has  been 
allowed  to  generate  a  current  for  some  time.  The  resulting 
current  at  once  indicates  that  the  former  has  the  greater  elec- 
tromotive force. 

XII.     DETERMINATION   OF   RESISTANCE. 

Numerous  experiments  shew  that  when  two  points,  which  are 
kept  at  a  constant  difference  of  potential,  are  connected  by  a 
conductor,  the  intensity  of  the  current  which  passes  depends 
on  the  dimensions  of  the  conductor,  and  on  the  substance  of 
which  it  is  composed. 

If  the  conductor  is  prismatic  in  form,  and  of  uniform  cross- 
section,  the  strength  of  the  current  varies  inversely  as  the 
length  and  directly  as  the  cross-section.  Further,  if  different 
currents  are  made  to  pass  through  the  same  conductor  by  vary- 
ing the  potential  .difference  of  its  terminals,  the  ratio  of  the 
current  strength  to  the  corresponding  potential  difference  is 
within  wide  limits  a  constant  quantity.  This  constant  is  termed 
the  resistance  of  the  conductor. 

These  statements  which  represent  the  laws  governing  the 
flow  of  electric  currents  are  summarized  in  the  relation  called 

Ohm's  Law,  C  =  — >  where  C  is  the  current  intensity,  E  the 
R 

potential  difference  of  the  terminals  of  the  conductor,  and  R  its 
resistance,  which  when  the  conductor  is  homogeneous  is  equal 

to  s — >  /  being  the  length  of  the  conductor,  A  its  cross-section, 

A 

and  s  a  constant  called  the  specific  resistance  of  the  substance 
composing  it. 


240 


EXPERIMENTAL   PHYSICS. 


A  conductor  therefore  possesses  an  absolute  unit  of  resist- 
ance when  a  unit  difference  of  potential  between  its  terminals 
causes  a  current  of  unit  strength  to  pass  through  it.  This  unit 
being  inconveniently  small  for  practical  purposes,  that  chosen  is 
the  ohm,  which  is  equal  to  io9  absolute  units  in  the  C.  G.  S.  sys- 
tem, and  is  the  resistance  of  a  conductor  when  a  difference  of 
potential  of  one  volt  between  its  ends  causes  a  current  of  one 
ampere  to  flow  through  it. 

The  ohm  is  realized  practically  in  the  resistance  offered  to  a 
steady  electric  current  by  a  uniform  column  of  pure  mercury 
106.3  centimeters  long  and  one  square  millimeter  in  cross- 
section. 

Standard  ohms  are  usually  constructed  of  mercury,  or  of 
some  metal  whose  resistance  varies  but  little  with  slight  changes 
in  temperature. 

One  form  of  a  mercurial  standard  is  shewn  in  Fig.  35.  The 
mercury  is  contained  in  a  fine  capillary  tube,  bent  into  a  num- 
ber of  turns,  and  connected  to  large  open 
tubes,  also  partly  filled  with  mercury.  This 
capillary  tube  is  calibrated  with  extreme 
care,  and  in  determining  its  length  allowance 
is  made  for  the  resistance  of  the  mercury 
in  the  large  tubes.  When  in  use,  it  is  sus- 
pended as  shewn  in  the  figure,  in  a  glass 
vessel  filled  with  pieces  of  melted  ice. 

These  standards  are  extremely  fragile,  and 
it  is  customary  in  laboratory  practice  to 
replace  them  by  those  made  of  metal  wire. 
The  wire,  Fig.  36,  is  coiled  in  a  double  spiral, 
and  its  ends  being  attached  to  two  bent  cop- 
per rods  A,  B,  it  is  then  placed  between  two 
brass  cylinders  E  and  C,  and  the  intervening 
Fig.  35-  space  is  filled  with  paraffine.  When  the 

standard  is  being  used,  the  free  ends  of  the 
rods    A,   B   are    placed   in   mercury  cups,  which  form  part  of 


DETERiMINATION    OF    RESISTANCE. 


241 


the  circuit,  the  cylinder  C  is  partly  immersed  in  a  water  bath, 
and  the  temperature  of  the  wire  is  ascertained  by  means  of  a 
thermometer  inserted  in  the  cylindrical  opening  D.  The  tem- 


Fig.  36. 

perature  at  which  the  standard  is  accurate  is  usually  inscribed 
on  it,  and  having  been  thoroughly  tested,  the  coil  is  accom- 
panied by  a  statement  shewing  its  variations,  with  changes  of 


Fig.  37. 

temperature.     In  order  that  these  may  be    small   the  wire   is 
usually  made  from  an  alloy  of  platinum  and  silver. 

The  resistance  of  a  conductor  is  determined  by  comparison 


242 


EXPERIMENTAL    PHYSICS. 


with  others  whose  resistances  are  known.  •  These  which  are  ac- 
curate copies  of  standards  are  generally  made  in  coils,  and  are 
arranged  conveniently  in  boxes  provided  with  some  device  for 
rapidly  throwing  one  or  more  of  them  in  or  out  of  circuit.  Such 
an  arrangement  is  exhibited  in  Fig.  37,  and  by  reference  to  Fig. 
38,  the  manner  in  which  the  coils  are  inserted  may  be  readily 
seen.  The  wire  corresponding  to  any  particular  resistance  is 
bent  double  at  its  middle  point,  and  is  wound  as  a  double  spiral 
(to  prevent  induction)  about  either  one  or  two  bobbins,  such  as 
A  and  B.  These  are  attached  to  a  plate  of  ebonite,  which  forms 
the  cover  of  the  box,  by  means  of  two  large  brass  or  copper 
screws,  which  pass  completely  through  the  ebonite,  and  have 


their  ends  firmly  embedded  in  two  brass  plates,  C  and  D.  These 
plates,  which  form  part  of  the  circuit,  are  connected  electrically 
by  soldering  an  end  of  the  spiral  to  each  of  the  screws.  Plugs 
of  brass,  such  as  E,  are  made  to  fit  snugly  into  grooves  provided 
for  them  between  the  brass  plates.  When  these  are  inserted 
the  current 'passes  directly  from  one  plate  to  the  other  without 
going  through  the  coil,  since  the  plugs  offer  practically  no  resist- 
ance to  its  passage.  In  order,  therefore,  to  throw  any  particu- 
lar resistance  into  circuit  it  is  only  necessary  to  remove  the 
corresponding  plug.  In  the  manufacture  of  these  coils  German 
silver  is  extensively  used,  but  for  very  high  resistances  they  are 
generally  made  from  an  alloy  of  platinum  and  silver. 


DETERMINATION    OF    RESISTANCE. 


243 


Resistances  of  Solids.  —  METHOD  I. — By  substitution. 

In  order  to  determine  the  resistance  in  question,  it  is  placed 
in  the  same  circuit  as  a  sensitive  galvanometer,  and  a  cell 
or  battery  of  constant  electromotive  force.  When  the  current 
has  become  steady,  the  deflection  obtained  is  noted.  The  un- 
known resistance  is  then  replaced  by  a  box  of  coils,  and  by  with- 
drawing the  plugs  resistances  are  thrown  in  circuit  sufficient 
to  reestablish  the  original  intensity  of  the  current,  the  sum 
of  those  so  inserted  being  equal  to  that  of  the  unknown. 

The  method  is  simple  but  primitive,  and  the  results  obtained 
are  only  approximate.  It  is  difficult  to  obtain  a  cell  which  will 
remain  constant  during  the  experiment,  and  the  resistance 
boxes  ordinarily  used  are  not  graded  with  sufficient  delicacy 
to  be  used  in  this  way. 

METHOD  II.  — By  ^lse  of  a  differential  galvanometer. 

In  a  differential  galvanometer  there  are  two  distinct  coils 
of  equal  resistance,  which  are  so  wound  that  when  currents  of 
equal  intensity  are  passed  through  them  in  opposite  directions 
no  effect  is  produced  on  the  needle.  In  determining  the  re- 
sistance of  a  conductor,  it  is  placed  in  circuit  .with  one  of  these 
coils,  and  a  rheostat  or  resistance  box  with  the  other.  The 
current  from  a  battery  is  made  to  divide  so  that  part  goes 
through  one  coil  in  one  direction,  and  the  remainder  in  the 
opposite  direction  through  the  other. 

The  needle  is  then  deflected,  and  suitable  resistances  are 
thrown  in  circuit  by  means  of  the  resistance  box  to  bring  the 
needle  back  to  its  initial  position.  The  sum  of  the  resistances 
so  inserted  is  in  this  case  also  equal  to  that  of  the  conductor 
tested.  This  method  possesses  the  advantage  of  being  sensi- 
tive and  also  of  being  independent  of  all  variations  in  the  inten- 
sity of  the  current,  but  it  is  difficult  to  construct  a  galvanometer 
which  will  have  the  properties  assumed. 

The  accuracy  of  the  instrument  can  be  tested  by  noting 
whether  there  is  any  deflection  of  the  needle  when  the  same 


244 


EXPERIMENTAL   PHYSICS. 


current  is  passed  through  the  galvanometer  by  one  coil  and 
back  by  the  other,  and  also  when  any  given  current  is  allowed 
to  divide  freely  between  the  two  coils  when  their  corresponding 
terminals  are  connected. 

METHOD  III.  —  By  use  of  a  tangent  galvanometer. 

A  circuit  is  made  so  as  to  include  a  constant  battery,  a  finely 
graduated  tangent  galvanometer,  a  rheostat,  and  a  known  resist- 
ance r,  approximately  equal  to  that  of  the  unknown. 

By  means  of  the  rheostat  the  intensity  of  the  current  is 
modified  so  that  the  needle  comes  to  rest,  making  an  angle 
of  about  45°  with  the  meridian.  The  unknown  resistance  x 
is  then  added  to  the  circuit,  and  when  the  needle  again  comes 
to  rest  its  deflection  is  noted.  Finally,  a  reading  is  taken  when 
both  r  and  x  are  removed  from  the  circuit.  If  a,  /3,  and  7  are 
the  deflections  in  these  cases  respectively,  we  have, 
E 


G  +  r 
E 


(i) 
(2) 


G  +  r+x 

|  =  ^tan7,  (3) 

where  E  is  the  electromotive  force  of  the  battery,  and  G  the 
resistance  of  the  whole  circuit  exclusive  of  r  and  x. 
From  (i),  (2),  and  (3)  the  relation 

r+x _tan«     tan  7  — tan  /3 
r       tan/3    tan  7  — tana 

is  obtained,  and  from  it  x  can  be  calculated. 

METHOD  IV.  —  The  Wheatstone  bridge. 

While  the  methods  previously  given  are  of  importance  in 
special  cases,  that  generally  adopted  in  laboratory  practice 
is  the  method  of  the  Wheatstone  bridge.  Its  sensitiveness  is 
limited  only  by  that  of  the  galvanometer,  and  it  excels  in  admit- 
ting of  rapid  and  extremely  accurate  determinations. 


DETERMINATION    OF    RESISTANCE. 


245 


The  principle  of  the  method  is  illustrated  in  Fig.  39. 

The  main  circuit  of  a  battery  E  divides  at  A  into  two 
branches,  ABC  and  ADC,  which  contain  the  resistances  a,  x 
and  b,  c,  respectively.  Between  the  terminals  of  these  resist- 
ances B  and  D  a  galvanometer  G  is  inserted,  and  as  a  current 
of  short  duration  only  is  required,  both  this  circuit  and  the 
main  one  contain  contact  keys  to  permit  the  current  to  be 
established  or  broken  at  will. 

In  making  a  test  the  unknown  resistance  x  is  inserted  between 
B  and  C,  and  the  other  resistances  a,  b,  and  c  are  so  adjusted 


Fig.  39. 

that  on  depressing  the  key  L,  no  current  passes  through  the 
galvanometer.  The  points  B  and  D  are  then  at  the  same 
potential.  Denoting  it  by  Vv  and  the  potentials  of  the  points 
A  and  C  by  V±  and  Vz  respectively,  it  follows  from  Ohm's  law, 
since  the  intensity  of  the  current  in  each  of  the  circuits  ABC 
and  ADC  is  the  same  throughout  its  length,  that 


and 


(i) 
(2) 


246 


EXPERIMENTAL   PHYSICS. 


Dividing  each  member  of  (i)  by  the  corresponding  one  of  (2), 
we  obtain  the  relation 


ax 

-=-  or 
be 


If,  therefore,  the  resistance  of  a  or  c,  and  either  the  resist- 
ances of  the  other  two,  or  their  ratio,  be  known,  that  of  x  can 
be  calculated.  It  is  best  to  take  b  equal  to  a  at  first,  and  then 
to  vary  c  until  by  trial  it  is  approximately  equal  to  x.  By 


Fig.  40. 

then  giving  a  a  certain  value,  and  gradually  increasing  b,  that 
of  x  can  be  determined  with  a  degree  of  accuracy  which  is 
limited  only  by  the  resistances  available,  and  as  already  indi- 
cated by  the  sensitiveness  of  the  galvanometer. 

A  practical  form  of  the  bridge  consists  of  a  resistance  box 
containing  three  sets  of  coils.  One  of  the  best  arrangements 
is  that  shewn  in  Fig.  40,  where  a  and  b  of  the  theoretical 
diagram  are  represented  by  two  sets  of  ten,  one  hundred,  and 
one  thousand  ohm  coils.  These  are  inserted  between  the  brass 


DETERMINATION    OF    RESISTANCE. 


247 


plates  bearing  these  numbers,  and  the  two  brass  strips  shewn 
in  the  figure  connected  to  A  and  H.  In  order  to  place  any 
one  of  them  in  circuit  it  is  only  necessary  to  insert  a  plug  in 
the  corresponding  aperture.  The  third  resistance  c  is  given  a 
wider  range,  and  consists  of  thirty-six  coils  placed  between  B 
and  G,  and  graded  in  powers  of  ten  so  as  to  give  any  resistance 
from  i  to  9999  ohms. 

The  unknown  resistance  is  placed  between  A  and  B,  the 
galvanometer  between  F  and  E,  and  the  battery  between  C 
and  D.  The  two  keys,  K'  and  K,  correspond  to  those  in 
Fig.  39  indicated  by  K  and  L  respectively,  and  the  dotted 
lines  exhibit  the  paths  of  the  current  through  the  instrument. 

Besides  being  exceedingly  compact,  this  form  of  bridge 
possesses  the  advantages  of  having  a  minimum  number  of 
plugs,  and  of  permitting  readings  to  be  made  directly.  In 
determining  a  resistance  care  should  be  taken  to  depress 
the  battery  key  K'  before  that  of  the  galvanometer  K,  and  to 
release  them  in  the  reverse  order,  else  the  disturbing  effects 
of  induction  currents  will  be  experienced  and  unnecessary  delay 
be  caused. 

When  a  preliminary  test  is  being  made  the  galvanometer  key 
should  not  be  pressed  down  firmly,  otherwise  the  contact  will 
be  good,  and  the  galvanometer  needle  may  be  so  violently 
deflected  as  to  break  the  suspending  fiber.  On  account  of  its 
constant  electromotive  force,  a  Daniell  cell  is  peculiarly  suited 
for  work  on  the  Wheatstone  bridge.  If,  however,  such  a  cell 
is  not  available,  others  may  be  used,  and  in  case  the  current  is 
too  strong  part  of  it  may  be  short-circuited  by  the  insertion 
of  a  shunt. 

In  comparing  the  conducting  powers  of  different  materials  it 
is  necessary  to  know  their  specific  resistances.  In  the  case  of 
metals,  or  other  solids,  these  can  be  readily  determined  for  any 

temperature  by  finding  the  resistances  of  long  uniform  wires  or 

jt 
rods  of  the  substances,  and  then  applying  the  relation  s  =  R  — 


248  EXPERIMENTAL   PHYSICS. 

Resistances  of  Liquids.  —  In  determining  the  specific  resist- 
ance of  a  liquid  such  as  mercury  a  capillary  tube  is  attached  by 
rubber  tips  to  two  small  steel  or  wooden  cups  which  have  short 
tubes  protruding  from  their  sides. 

The  liquid  in  one  of  these  cups  is  forced  through  the  capil- 
lary tube,  expelling  the  air,  and  is  made  to  gradually  fill  up  the 
cup  to  which  the  other  end  is  attached. 

Terminal  wires  are  then  inserted  in  the  mercury  in  the  cups, 
and  the  resistance,  which  will  be  practically  that  of  the  liquid  in 
the  capillary  tube,  is  determined  by  one  of  the  methods  given 
above.  It  only  remains  to  calculate  the  dimensions  of  this  tube 
and  to  apply  the  relation  already  given. 

For  very  accurate  results  the  tube  must  be  carefully  cali- 
brated by  the  method  adopted  in  the  construction  of  ther- 
mometers, but  for  ordinary  purposes  that  explained  on  page  32 
may  be  followed. 

In  order  to  insure  a  uniform  temperature  for  making  the  test, 
the  cups  and  the  capillary  tube  should  be  placed  in  a  water  or 
oil  bath. 

In  case  the  liquid  is  an  electrolyte,  a  serious  difficulty  is  met 
with  in  the  polarization  of  the  electrodes.  On  passing  a  current 
through  such  a  liquid,  its  resistance  is  altered  owing  to  changes 
in  its  condition  near  the  electrodes,  and  to  the  presence  of  the 
chemical  products  deposited  there. 

The  existence  of  an  electromotive  force  opposed  to  that  pro- 
ducing the  current  still  further  complicates  the  problem,  and 
renders  an  accurate  determination  of  the  resistance  difficult. 

It  has  been  found  when  the  liquid  examined  is  a  solution  of  a 
metallic  salt,  that  if  the  electrodes  are  composed  of  the  metal 
forming  the  base  of  this  salt  the  polarizing  effects  are  small, 
and  the  resistance  of  the  liquid  may  be  found  approximately  by 
the  ordinary  methods. 

One  of  the  early  devices  adopted  was  to  place  the  liquid  in  a 
rectangular  trough,  and  to  have  movable  electrodes  inserted  in 
it.  The  resistance  of  a  certain  length  of  the  liquid  was  first 


TEMPERATURE   COEFFICIENT    OF    RESISTANCE. 


249 


determined,  and  then  the  distance  between  the  electrodes  being 
shortened  by  a  known  length,  the  current  was  restored  to  its 
original  intensity  by  means  of  a  rheostat  placed  in  the  circuit. 
The  amount  of  the  wire  resistance  thus  added  was  taken  as  a 
measure  of  the  resistance  of  the  liquid  through  which  the 
electrode  was  moved. 

Alternating  currents  have  been  applied  with  considerable 
success  to  finding  the  resistances  of  electrolytes.  A  Wheat- 
stone  bridge  of  special  construction  is  used,  but  the  theoretical 
arrangements  are  the  same  as  have  already  been  described. 

An  electro-dynamometer,  however,  takes  the  place  of  the 
galvanometer,  and  this  instrument  again  is  sometimes  replaced 
with  advantage  by  a  telephone.  When  the  latter  is  used,  and 
the  alternating  currents  are  produced  by  means  of  a  rapidly 
vibrating  induction  coil,  a  humming  sound  is  heard,  which, 
when  the  proper  proportions  obtain  among  the  resistances, 
weakens  to  a  minimum  and  nearly  disappears. 

In  this  method  polarization  is  avoided  since  the  current  is 
passed  through  the  liquid  alternately  in  opposite  directions. 

XIII.   TEMPERATURE   COEFFICIENT   OF   RESISTANCE. 
SLIDE    WIRE    BRIDGE. 

The  resistance  of  a  conductor  depends  not  only  on  its  dimen- 
sions, and  on  the  substance  of  which  it  is  composed,  but  also 
on  its  temperature.  Owing  to  the  extensive  use  of  resistance 
coils  for  comparative  purposes,  it  becomes  therefore  a  matter  of 
great  importance  to  select  materials  for  their  construction  which 
vary  but  little  with  temperature,  and  to  know  exactly  the  law 
governing  such  variations. 

The  resistances  of  alloys  such  as  platinum  silver,  German 
silver,  gold  silver,  and  platinoid,  increase  much  less  rapidly  with 
a  rise  in  temperature  than  do  pure  metals,  while  carbon  forms  a 
notable  exception  to  the  general  law  for  elemental  substances 
in  that  its  resistance  decreases  as  its  temperature  rises. 


250  EXPERIMENTAL   PHYSICS. 

In  determining  a  law  for  this  variation  in  the  resistance  of 
metals  a  wire  of  suitable  length  of  the  substance  in  question 
is  selected,  and  a  series  of  experiments  are  performed  to  ascer- 
tain exactly  its  resistances  at  different  temperatures. 

By  means  of  the  results  obtained  the  law  is  then  exhibited 
graphically  by  plotting  a  curve,  taking  temperatures  for  abscissae, 
and  resistances  for  ordinates  ;  but  for  the  purposes  of  calcu- 
lation it  is  better  to  follow  the  method  of  Least  Squares,  and  to 
establish  an  empirical  formula  connecting  the  resistances  of  the 
conductor  with  its  corresponding  temperatures. 

This  relation  is  usually  obtained  in  the  form 


in  which  R0  is  the  resistance  of  the  wire  at  zero,  and  the 
coefficients  a  and  £  are  calculated  from  the  observed  results. 

As  these  coefficients  depend  only  on  the  substance  of  the 
wire,  this  relation  can  therefore  be  applied  to  ascertain  the  resist- 
ance of  any  conductor  of  the  same  material  at  any  given  tem- 
perature when  its  value  at  o°  C.  is  known.  In  order  to  arrive  at 
results  which  will  be  reliable,  every  precaution  must  be  taken 
to  measure  the  resistances  of  the  wire  accurately,  and  to  note 
its  temperatures  with  the  greatest  precision. 

The  wire,  which  should  be  well  insulated,  is  wound  on  a  thin 
hollow  bobbin  made  of  wood,  and  is  inserted  in  a  thin  glass 
tube  closed  at  one  end.  A  delicate  thermometer  is  also  placed 
in  this  tube,  and  the  whole  is  suspended  in  a  liquid  bath  which 
should  be  provided  with  a  stirrer,  and  a  second  thermometer  to 
check  the  readings  of  the  first. 

The  resistance  of  the  wire  is  generally  determined  by  means 
of  a  Wheatstone  bridge,  the  connections  being  made  by  solder- 
ing two  stout  wires  to  its  terminals. 

When  a  test  is  about  to  be  made  a  Bunsen  flame  is  applied 
to  the  bath  until  it  reaches  a  suitable  temperature.  It  is  then 
withdrawn,  and  the  liquid  is  stirred  until  the  two  thermometers 
shew  that  its  temperature  has  become  steady.  The  resistance 


TEMPERATURE   COEFFICIENT   OF    RESISTANCE. 


251 


is  then  determined,  and  the  corresponding  temperature  of  the 
wire  is  found  by  taking  the  mean  of  the  readings  of  the  two 
thermometers.  More  heat  is  then  applied,  and  the  same  opera- 
tions are  repeated  until  the  whole  range  of  temperatures  chosen 
has  been  covered. 

If  it  is  desired  to  make  this  somewhat  extensive,  an  oil  bath 
should  be  used,  but  as  it  emits  an  exceedingly  offensive  odour 
it  is  better  for  ordinary  purposes  to  use  water  instead  of  oil. 
To  increase  the  accuracy  of  the  work  the  test  should  be  con- 
ducted with  falling  as  well  as  rising  temperatures. 

The  errors  which  necessarily  accompany  the  results  obtained 
by  using  the  ordinary  form  of  the  Wheatstone  bridge  cannot  be 


Fig.  41. 

neglected  in  a  delicate  experiment  such  as  this,  and  a  simplified 
form  of  the  instrument  known  as  the  slide  wire  bridge  is  gen- 
erally used. 

In  connection  with  Fig.  39,  it  has  been  shewn  that  when  the 
points  B  and  D  are  at  the  same  potential,  the  resistances  are 

connected  by  the  relation  x  =  ^c.     In  the  slide  wire  bridge,  a 

b 

single  wire  of  homogeneous  structure  and  of  constant  cross- 
section  takes  the  place  of  the  resistances  b  and  c,  and  in 
determining  that  of  x,  the  galvanometer  terminal  is  moved 
along  this  wire  until  a  point  D  is  reached,  which  is  at  the 
same  potential  as  that  of  B. 


252 


EXPERIMENTAL   PHYSICS. 


Since  the  resistance  of  any  portion  of  this  wire  is  propor- 
tional to  its  length,  that  of  the  unknown  is  given  by  x  =  -j<z, 

*i 
where  /j  and  /2  are  respectively  the   lengths   AD  and   DC  of 

the  wire  AC. 

In  its  practical  form,  the  bridge  is  shewn  in  Fig.  41.  The 
wire,  which  may  be  of  brass  or  an  alloy  of  platinum,  is  one 
meter  long,  and  is  tightly  stretched  between  the  points  C  and 
K.  A  heavy  brass  bar,  on  which  is  ruled  a  meter  scale  with 
centimeter  divisions,  is  so  placed  in  front  of  this  that  the  ter- 
minals of  the  wire  are  opposite  the  limiting  divisions  of  the 
scale. 

A  slider  A  provided  with  a  vernier,  and  with  a  key  B  for 
making  contact  with  the  wire,  can  move  freely  along  this  bar, 
and  a  series  of  brass  strips,  shewn  in  the  figure,  supply  the 
remaining  connections. 

The  unknown  resistance  and  a  standard  are  introduced  into 
the  circuit  by  means  of  four  mercury  cups,  two  of  which  are 
of  brass  and  two  of  ebonite,  and  the  relative  positions  of  these 
can  be  reversed  by  means  of  the  commutator  D,  constructed 
of  two  U-shaped  brass  or  copper  rods,  whose  terminals  also 
rest  in  the  mercury  cups. 

The  battery  terminals  are  attached  to  the  bridge  at  E  and  F, 
and  those  of  the  galvanometer  to  the  binding  poles  G  and  //", 
which  are  connected  to  the  graduated  brass  bar  and  to  the 
central  metallic  strip  supporting  the  mercury  cups  respectively. 
The  slider  A  can  be  clamped  in  any  position,  and  the  key  is 
so  arranged  that  by  depressing  it  a  spring  is  acted  on  which 
causes  a  rounded  metallic  knife-edge  to  press  gently  but  uni- 
formly against  the  stretched  wire. 

Best  results  are  obtained  when  /x  and  /2  are  nearly  equal, 
and  the  unknown  resistance,  therefore,  should  be  always  so 
selected  that  it  is  at  o°  C.  approximately  equal  to  that  of  the 
standard. 

Care  must  be  taken  not  to  touch  the  wire  or  the  other  metal- 


GALVANOMETER    RESISTANCE. 


253 


lie  parts  of  the  bridge  by  the  hands  during  a  test,  otherwise 
thermo-electric  currents  are  at  once  set  up,  which  greatly  dis- 
turb the  readings. 

The  slide  wire  bridge  is  especially  useful  in  the  construction 
of  resistance  coils  which  are  intended  to  be  very  accurate 
copies  of  standards. 

XIV.  GALVANOMETER  RESISTANCE. 

If  the  laboratory  is  provided  with  a  second  galvanometer 
sufficiently  sensitive,  the  coils  of  the  one  to  be  examined  may 
be  treated  as  an  ordinary  conductor,  and  their  resistance  deter- 
mined by  one  of  the  methods  outlined  in  the  last  experiment. 

If,  however,  a  second  instrument  is  not  at  hand  or  is  not 
conveniently  adjusted,  either  the  method  of  equal  deflections 
or  that  known  as  Thomson's  may  be  applied,  and  in  these 
tests,  the  galvanometer  whose  resistance  is  required  is  itself 
used  as  the  current  indicator. 

(i)  Method  of  equal  deflections. 

The  arrangement  is  that  shewn  in  Fig.  42.  The  current 
which  is  supplied  by  a  constant  battery  E  divides  at  C,  part 


Fig.  42. 

going  through  the  galvanometer  G,  and  part  through  a  shunt 
5.     The  main  circuit  also  contains  a  variable  resistance  AB. 


254 


EXPERIMENTAL   PHYSICS. 


Denoting  the  resistance  of  the  galvanometer  by  g,  that  of  the 
shunt  by  s,  and  that  of  the  battery  and  leading  wires  by  r, 
we  have,  when  the  resistance  of  AB  is  Rlt  the  intensity  of 
the  main  current  given  by 

c        E 

L'  9  /     x 


and  that  of  the  part  passing  through  the  galvanometers  by 


w 


If  now  the  shunt  be  removed  from  the  circuit,  and  the  re- 
sistance of  AB  altered  until  the  galvanometer  indicates  that 
the  original  intensity  of  the  current  passing  through  it  is 
restored,  the  strength  of  the  current  is  given  by 


where  Rz  is  the  new  value  of  AB. 

Equating  the  right-hand  members  of  (2)  and  (3),  it  follows 
that  the  resistance  of  the  galvanometer  may  be  found  from  the 
relation, 


As  the  theory  indicates,  it  is  absolutely  necessary  for  a  suc- 
cessful application  of  this  method  to  use  only  a  battery  which 
will  remain  constant  for  a  considerable  time.  The  method  is 
somewhat  tedious,  owing  to  the  necessity  of  previously  deter- 
mining the  internal  resistance  of  the  battery,  and  best  results 
are  obtained  when  a  battery  is  selected  whose  resistance  is  so 
small  that  it  can  be  neglected. 


GALVANOMETER  RESISTANCE. 


255 


In  that  case,  if  the  resistance  of  the  leading  wires  is   also 
neglected,  that  of  the  galvanometer  is  given  by 


(5) 


(2)    Thomson  s  method. 

This  is  an  application  of  the  principle  of  the  Wheatstone 
bridge,  similar  to  that  described  in  Experiment  XII.  The  gal- 
vanometer is  disposed,  as  shewn  in  Fig.  43,  between  the  two 
points  C  and  D,  and  a  contact  key  is  placed  in  the  circuit 
between  C  and  B. 

If,  when  the  key  is  raised  and  a  current  is  passing  through 
the  galvanometer,  the  points  C  and  B  are  not  at  the  same 


potential,  then  a  current  will  pass  between  these  points  when 
the  key  is  depressed,  and  a  consequent  alteration  in  the  deflec- 
tion of  the  galvanometer  will  occur. 

If,  however,  the  resistances  a,  b,  and  c  are  so  adjusted  that 
C  and  B  are  at  the  same  potential,  it  is  quite  immaterial 
whether  the  key  is  raised  or  depressed,  as  no  current  will  pass 
between  these  points,  and  there  will  therefore  be  no  change 


256  EXPERIMENTAL   PHYSICS. 

in   the   intensity  of   the    current    passing   through   the   galva- 
nometer. 

When  this  condition  obtains  the  ordinary  relation  for  the 
equilibrium  of  the  Wheatstone  bridge  applies,  and  the  resistance 
of  the  galvanometer  is  given  by 


The  method  is  exceedingly  simple,  and  is  preferable  to  the  for- 
mer, in  that  it  is  quite  independent  of  the  resistance  of  the  battery. 

In  case  the  battery  used  does  not  remain  constant  during 
the  experiment,  the  deflection  of  the  galvanometer  needle  will 
gradually  change.  This,  however,  will  not  affect  the  application 
of  the  method,  as  the  test  consists  entirely  in  noting  whether  the 
intensity  of  the  current  passing  through  the  galvanometer  is 
affected  by  raising  or  depressing  the  contact  key. 

In  case  the  current  through  the  galvanometer  produces  too 
great  a  deflection,  a  shunt  should  be  inserted  between  the  points 
A  and  D.  By  giving  different  values  to  this,  the  current  pass- 
ing through  the  bridge  may  then  be  suitably  modified. 

XV.     RESISTANCE   OF   BATTERIES   OR   CELLS. 

From  the  exercises  on  batteries  in  Experiment  XL,  it  is 
evident  that  a  cell  or  battery  acts  in  the  same  manner  as  an 
ordinary  conductor,  and  offers  a  resistance  to  the  passage  of  the 
current  which  it  produces. 

The  value  of  this  resistance  depends  to  some  extent  on  the 
electrodes  of  the  cell,  on  the  amount  of  their  surfaces  submerged, 
and  on  the  distance  they  are  apart.  It  also  varies  within  very 
wide  limits  with  the  composition  of  the  battery  solutions,  and 
with  their  concentration. 

It,  however,  cannot  be  considered  a  definite  quantity,  since 
polarization  occurs  in  most  cells  as  soon  as  the  current  is  estab- 
lished, and  the  resistance  gradually  increases,  its  value  at  any 
instant  depending  both  on  the  strength  of  the  current  then 


RESISTANCE   OF   BATTERIES   OR   CELLS.  257 

passing  and  on  the  length  of  time  the  circuit  has  been  closed. 
In  determining  the  value  of  this  quantity,  therefore,  for  a  cur- 
rent of  given  intensity,  the  result  obtained  can  be  taken  only  as 
an  approximation  to  its  correct  value  under  other  circumstances  ; 
but  such  determinations,  however,  are  sufficient  for  ordinary 
purposes,  and  may  be  made  by  one  or  other  of  the  following 
methods  : 

(i)   Ohms  method. 

This  method  is  a  direct  application  of  Ohm's  law  to  a  com- 
plete circuit,  including  the  battery  to  be  tested,  a  sensitive 
galvanometer,  and  a  variable  resistance  R. 

Denoting  the  electromotive  force  of  the  cell  by  E,  the  resist- 
ance of  the  galvanometer  and  leading  wires  by  g,  and  that  of 
the  cell  by  r,  the  relation 


will  represent  the  intensity  of  the  current  produced  when  Rl  is 
the  resistance  given  to  R.     On  changing  the  variable  resistance 


to  Rv  a  second  relation 


is  obtained  ;  and,  assuming  that  E  remains  constant  during  the 
test,  it  follows  from  (i)  and  (2)  that 


If  the  readings  of  the  galvanometer  follow  the  tangent  law, 
this  relation  becomes 


_  pr 


_ 

tan  #!-  tan  02 


in  which  dl  and  02  are  the  deflections  corresponding  to  the  cur- 
rents C^  and  Cz. 


258 


EXPERIMENTAL   PHYSICS. 


In  adopting  this  method  a  number  of  readings  should  be 
taken  by  varying  the  resistance  R,  and  in  each  case  the  results 
should  be  checked  by  reversing  the  current  through  the  galva- 
nometer. 

(2)    Thomsons  method. 

As  shewn  in  Fig.  44,  the  battery  E,  whose  resistance  is 
required,  is  joined  in  series  with  a  galvanometer  G  and  a 
variable  resistance  AB,  while  a  shunt  5  is  inserted  between 
the  points  F  and  H. 

In  applying  the  method  a  suitable  deflection  is  first  given  to 
the  galvanometer  needle  by  properly  adjusting  the  resistances 
AB  and  5.  The  shunt  is  then  removed,  and  the  rheostat 


IHitHi 


Fig.  44. 

altered,    until   the   original   intensity    of    the   current    passing 
through  the  galvanometer  is  reestablished. 

If  in  the  first  operation  s  and  Rl  are  the  values  given  to  vS 
and  to  AB,  and  in  the  second  R2  that  given  to  the  rheostat, 
the  expressions  for  the  current  passing  through  G  in  the  two 
cases  are, 


and 


(2) 


RESISTANCE    OF    BATTERIES    OR   CELLS. 


259 


By  equating  the  right-hand  members  of  these  equations,  the 
relation  r=^-f£—. — ^  may  be  deduced,  and  from  it  the  internal 


resistance  of  the  battery  may  be  calculated. 

(3)  Mances  method. 

In  applying  the  previous  methods  considerable  time  is  re- 
quired to  make  the  two  adjustments,  and  unless  the  element 
examined  polarizes  very  slowly,  the  results  obtained  cannot  be 
taken  *as  accurate. 

In  Mance's  test,  however,  the  time  which  elapses  between 
the  two  observations  is  exceedingly  short,  being  merely  that 
spent  in  closing  a  circuit,  and  it  is  therefore  not  sufficient  for 
the  battery  resistance  to  undergo  any  perceptible  change.  For 


this  reason  this  method  is  generally  followed  in  determining 
the  resistance  of  a  cell,  or  battery,  which  polarizes  very  rapidly. 
The  arrangement  is  that  shewn  in  Fig.  45.  There  the  cell 
is  inserted  in  one  of  the  arms  of  the  bridge,  the  galvanometer 
in  one  of  its  diagonals,  and  a  coarse  wire  of  very  small  resist- 
ance, containing  a  contact  key  in  the  other. 


26o  EXPERIMENTAL   PHYSICS. 

The  test  consists  simply  in  so  adjusting  the  resistances  a,  b, 
and  c  that  no  difference  is  observed  in  the  strength  of  the 
current  passing  through  the  galvanometer  when  the  key  is 
raised  or  depressed. 

When  the  resistances  are  so  disposed  that  the  proper  con- 

ditions obtain,  they  are  connected  by  the  relation  r  =7  c,  where 

b 

r  is  the  resistance  of  the  cell. 

This  may  be  shewn  by  considering  the  currents  passing 
through  the  galvanometer  under  the  two  arrangements.  .In  the 
first  the  current  divides  at  B,  and  takes  the  paths  BAG  and 
BC  to  reach  the  point  C.  The  strength  of  the  current  in 
this  case  is  given  by 

E 


In  the  second,  depressing  the  key  amounts  to  the  same  thing 
as  bringing  the  two  points  D  and  A  together.  Denoting  this 
hypothetical  point  of  union  by  DA,  the  current  may  be  con- 
sidered as  dividing  at  B,  and  then  passing  partly  by  BA  to 
DA,  and  partly  by  BC,  and  its  subdivisions  CD  and  CA  to  the 
same  point,  and  thence  back  to  the  battery. 

Under  these  circumstances  the  current  intensity  is  given  by 


be 


If,  then,  Ci  is  equal  to  Cv  it  will  be  found  on  equating  the 
right-hand  members  of  (i)  and  (2)  that  the  resistances  satisfy 
the  relation 

§        ~f<- 

This  can  also  be  readily  seen  by  considering  the  intensities 
of  the  currents'  in  the  various  branches  of  the  bridge  when  the 


DETERMINATION   OF   ELECTROMOTIVE   FORCES.       26  1 

key  is  depressed,  as  being  the  resultant  of  two  steady  systems, 
the  first  being  that  which  exists  when  the  key  is  raised,  and  the 
second  being  some  system  in  which  the  points  B  and  C  are  at 
the  same  potential.  As  is  evident  from  Method  IV.,  Experi- 
ment XII.,  this  can  be  considered  to  be  an  arrangement  in 
which  an  electromotive  force  is  inserted  in  the  circuit  AKD, 
and  in  which  consequently  the  resistances  of  the  arms  of  the 
bridge  are  connected  by  the  formula 

rrf* 

In  applying  this  method  it  will  be  found  that  in  most  cases 
the  galvanometer  deflection  will  be  inconveniently  large.  This 
defect  may  be  remedied  by  shunting  the  galvanometer,  or  if 
the  instrument  is  provided  with  a  directing  magnet,  by  so  dis- 
posing it  that  it  will  act  in  an  opposite  direction  to  that  of 
the  current,  and  so  reduce  the  deflection. 

If  the  laboratory  is  provided  with  a  potential  galvanometer, 
or  finely  graduated  voltmeter,  the  resistance  of  a  cell  may  be 
determined  by  observing  the  differences  of  potential  between 
the  terminals  of  the  element  first  in  open  circuit,  and  then 
when  the  circuit  is  completed  by  a  small  known  resistance  R. 
Denoting  these  differences  of  potential  by  E  and  V  respectively, 
it  follows  from  Ohm's  law  that 

E    =  V 
r+R     R 


or  that 


DETERMINATION  OF  ELECTROMOTIVE  FORCES. 

In  Experiment  XL  a  few  exercises  are  indicated  as  being 
suitable  for  giving  clear  and  accurate  notions  regarding  the 
electromotive  force  of  a  cell  or  battery.  The  following  article 
is  devoted  to  a  description  of  some  of  the  best  methods  that  may 
be  followed  in  determining  this  constant  for  a  given  element. 


262  EXPERIMENTAL    PHYSICS. 

As  already  stated,  the  electromotive  force  of  a  cell  is  meas- 
ured by  the  greatest  difference  of  potential  between  its  terminals 
in  open  circuit.  The  unit  potential  difference  in  the  C.  G.  S. 
system  is  inconveniently  small  for  practical  purposes,  and  that 
generally  adopted  is  the  volt.  It  is  equal  to  io8  units  in  the 
C.  G.  S.  system,  and  is  the  difference  of  potential  that  must  be 
maintained  between  the  terminals  of  a  conductor  of  one  ohm 
resistance  in  order  that  the  intensity  of  the  current  produced 
in  it  may  be  one  ampere. 

While  the  first  three  methods  are  for  comparative  determina- 
tions by  reference  to  some  standard  cell  whose  constant  is 
known,  the  following  ones  indicate  how  an  absolute  measure  of 
the  electromotive  force  of  an  element  may  be  obtained  in  terms 
of  resistance  and  current  intensity. 

I.    Comparative  determination  of  electromotive  forces. 

(i)    W/teatstones  method. 

A  standard  cell,  such  as  that  of  Daniell  or  Clark,  is  joined  in 
simple  circuit  with  a  galvanometer,  and  a  rheostat,  or  resistance 
box.  The  latter  is  adjusted  to  produce  a  convenient  deflection 
of  0°,  and  then  a  known  resistance  r  is  added  to  the  circuit,  and 
a  new  deflection  of  6^  is  obtained. 

The  standard  is  next  replaced  in  the  circuit  by  the  cell,  whose 
electromotive  force  is  required,  and  by  means  of  the  rheostat 
the  current  is  modified  until  the  initial  deflection  of  6°  is  repro- 
duced. It  only  remains  then  to  determine  what  resistance  1\ 
must  now  be  added  to  reestablish  the  second  deflection  of  0^, 
and  from  these  known  quantities  the  unknown  electromotive 
force  -£*!  can  be  calculated  in  terms  of  that  of  the  standard  E. 

Denoting  the  resistances  of  the  whole  circuit  in  the  two  cases, 
corresponding  to  the  deflection  of  6°  by  R  and  RI}  it  follows 
from  Ohm's  law,  that 

<•> 
andthat 


DETERMINATION    OF   ELECTROMOTIVE   FORCES.       263 
By  combining  these  we  have, 


and  therefore  the  electromotive  force  E1  is  -1  times  that  of  the 
standard. 

The  method  is  simple,  and  is  very  accurate  if  the  cells  used 
are  constant.  If  not,  however,  the  method  cannot  be  success- 
fully applied. 

(2)  Lnmsden  or  Lecoine's  method. 

The  arrangement  for  this  method  is  shewn  in  Fig.  46.  The 
standard  A  and  the  cell  to  be  tested,  B,  are  mounted  in  series 


i 


1 


Fig.  46. 

with  their  opposite  terminals  connected  in  a  circuit,  which  also 
contains  the  variable  resistances  E  and  F. 

A  shunt,  which  includes  a  sensitive  galvanometer,  connects 
the  points  C  and  D. 

In  applying  this  method  a  certain  value  is  given  to  one  of  the 
adjustable  resistances,  and  then  the  other  is  varied  until  no 
current  passes  through  the  galvanometer.  If  E  and  E±  denote 
the  electromotive  forces  of  the  cells  A  and  B  respectively,  and 
r  and  ^  their  internal  resistances,  these  quantities  are  connected 
by  the  relation 

§=§^7-  W 


264  EXPERIMENTAL  PHYSICS. 

R  and  R1  being  the  resistances  of  E  and  F  respectively  (in- 
cluding that  of  their  connecting  wires).  This  will  be  readily 
seen  from  a  consideration  of  the  following  theory  :  In  all  such 
cases  as  this  where  the  circuit  has  many  branches,  and  contains 
a  number  of  electromotive  forces,  the  principle  of  superposition 
is  to  be  applied.  Each  electromotive  force  in  combination  with 
the  others  has  precisely  the  same  effect  as  it  would  have  were 
it  the  only  one  in  the  circuit,  and  hence  the  problem  of  deter- 
mining the  current  passing  in  any  one  of  the  branches  can  be 
greatly  simplified  by  considering  the  action  of  each  of  the  cells 
separately. 

In  the  above  arrangement  the  two  cells  tend  to  send  a  cur- 
rent through  the  galvanometer  circuit  in  opposite  directions,  and 
since  no  deflection  is  produced  in  this  instrument,  the  currents 
which  they  would  send  through,  were  they  acting  separately, 
must  then  be  equal. 

That  which  would  pass  through  it,  were  the  standard  the  only 
one  acting,  is  given  by 

E  r  +  R      • 


r\  R  | 


while  if  the  cell  B  were  the  only  one  producing  a  current,  the 
amount  which  it  would  send  through  the  galvanometer  is  given 
by 

£1  _         r+R 

l+^+*) 

1    g+r+R 
If,  then,  C  is  equal  to  Cv  it  follows  that 


or  that 

Z7  .,      i     T? 

(3) 

If,  now,  r  and  ^  are  known,  E±  can  be  at  once  found  in  terms 
of  E,  but  if  not,  a  second  observation  is  made  by  increasing  R 


DETERMINATION    OF    ELECTROMOTIVE    FORCES. 


265 


by  some  value  p,  and  then  determining  what  resistance  must  be 
added  to  R±  to  again  reduce  the  current  in  CD  to  zero.  Denot- 
ing this  by  plt  we  have, 


E       r+R+p  ' 
and  combining  (3)  and  (4),  it  follows  that 


This  method,  just  as  the  preceding  one,  can  only  be  ap- 
plied successfully  to  constant  elements.  If,  however,  one  of 
the  cells  is  constant,  it  affords  a  means  of  studying  the  proc- 
ess of  polarization  in  the  other.  The  special  advantage  of 
the  method  is  that  both  cells  are  working  under  exactly  the 
same  conditions. 

(3)  Poggendorjfs  method. 

This,  which  is  probably  the  most  exact  of  all  the  methods 
devised,  involves  a  new  idea,  that  of  determining  the  electro- 
motive force  of  an  element  by  balancing  it  against  a  potential 
difference  maintained  in  a  circuit  by  means  of  a  standard  cell. 

-.  -  Hi  - 


K 


Fig.  47. 


As  Fig.  47  indicates,  the  standard  A  is  joined  in  series  with 
a  rheostat  F,  and  two  carefully  constructed  sliding  resistances 
L  and  M,  B  the  cell  to  be  tested  together  with  a  galvanometer 


266  EXPERIMENTAL   PHYSICS. 

G,  being  inserted  as  a  shunt  between  the  points  C  and  D  in  such 
a  manner  that  its  current  tends  to  oppose  that  produced  by  the 
standard  cell.  Contact  keys  are  provided  at  K  and  H,  and  the 
various  connections  of  the  circuit  are  made  of  stout  brass  or 
copper  strips,  in  order  that  their  resistance  may  be  negligible. 

When  the  arrangements  described  have  been  made  the  rheo- 
stat F  is  varied  until  on  depressing  the  contact  keys  the  galva- 
nometer shews  that  no  current  is  passing  along  CD. 

This  condition  implies  that  a  difference  of  potential  exists 
between  the  points  C  and  D  exactly  equal  to  the  electromotive 
force  of  the  cell  B. 

The  accuracy  of  this  adjustment  can  be  tested  by  first  giving 
the  rheostat  F  a  value  slightly  above  that  which  it  then  has, 
and  afterwards  one  slightly  below  it.  This  should  result  in  the 
galvanometer  indicating  a  current  passing  first  in  one  direction, 
and  then  in  the  other  along  CD. 

After  this,  a  known  resistance  R  is  thrown  in  circuit  by 
means  of  the  sliding  resistance  M,  and  then  a  corresponding 
amount  Rl  is  added  at  L  to  reproduce  the  condition  of  zero 
current  in  CD. 

The  electromotive  force  of  the  element  B  is  then  given  in 
terms  of  that  of  the  standard  E  by  the  relation 

EI=E      R 


This  follows  from  first  applying  Ohm's  law  to  the  whole 
circuit  ACKD,  and  then  to  the  part  CKD.  Denoting  the 
potential  difference  between  C  and  D  by  V^  —  F^,  the  resist- 
ance of  F  by  p,  and  that  of  the  battery  A  by  r,  we  have,  for 
the  first  adjustment, 

E       V  —  V,, 

and  for  the  second, 

r+p  +  R  +  R~  p+R 


DETERMINATION   OF   ELECTROMOTIVE   FORCES.       267 
Combining  (i)  and  (2),  we  have, 

''R  +  Ri 

and  since  the  potential  difference  V-^—  V^  is  equal  to  the  re- 
quired electromotive  force  El  the  relation  given  previously  is 
evident. 

Owing  to  the  determination  being  made  with  the  cell  practi- 
cally in  open  circuit,  errors  due  to  polarization  cannot  arise,  and 
the  method  can  therefore  be  readily  applied  to  all  classes  of 
cells,  including  those  which  polarize  even  very  rapidly.  The 
standard  selected  should  be  a  constant  element,  but  even  should 
it  vary,  errors  due  to  this  cause  may  be  avoided  by  closing  the 
circuit  only  for  short  intervals  at  each  test.  In  case  the  galva- 
nometer used  is  a  reflecting  one,  it  will  suffice  to  merely  depress 
the  keys  K  and  H  for  an  instant.  If  the  cell  tested  is  not 
properly  balanced,  the  momentary  contact  will  be  sufficient  to 
indicate  this. 

Like  the  preceding,  this  method  is  well  adapted  for  studying 
the  effects  of  polarization  in  a  cell  by  measuring  its  electro- 
motive force  after  it  has  been  allowed  to  produce  currents  of 
different  intensities  for  varying  intervals  of  time. 

As  the  theory  indicates,  the  electromotive  force  of  the  stand- 
ard must  be  higher  than  that  of  the  element  tested ;  if,  how- 
ever, it  is  lower,  the  method  can  still  be  applied  by  joining  a 
number  of  standard  cells  in  series  until  an  electromotive  force 
sufficiently  high  is  obtained. 

(4)   Clark's  method. 

This  method  is  based  on  the  same  principle  as  the  last,  but  it 
is  so  modified  that  the  standard  and  the  unknown  element  are 
compared  under  precisely  similar  circumstances. 

The  disposition  is  that  outlined  in  Fig.  48.  The  current  is 
produced  by  a  constant  battery  C,  of  resistance  r,  whose  circuit 
CDEF  contains  a  rheostat  M,  and  a  sliding  resistance  EF. 


268 


EXPERIMENTAL   PHYSICS. 


The  standard  A,  together  with  a  galvanometer,  is  inserted  as 
a  shunt  between  the  points  D  and  P,  and  B  the  cell  to  be 
examined  and  another  galvanometer  are  inserted  in  a  second 
derived  circuit  EKN. 

The  standard  cell  is  first  balanced  by  properly  adjusting  the 
resistance  of  the  rheostat,  and  then  the  contact  key  K  is 


Fig.  48. 

depressed  and  the  terminal  .V  is  moved  along  EF,  until  a 
point  is  reached  when  no  deflection  is  observed  in  the  gal- 
vanometer G. 

The  cells  A  and  B  are  then  both  balanced,  and  the  current  is 
produced  by  the  battery  C  alone. 

Denoting  the  electromotive  forces  of  A,  B,  and  C  by  E,  Elt 
and  Ez  respectively,  the  resistances  of  the  two  parts  of  EF  by 
R  and  RI}  and  that  of  the  rheostat  by  p,  we  have, 

(I) 
and  _A._,_  =  ^.  (2) 

The    unknown    electromotive   force    is    then    given    by   the 


relation, 


R 


.  E. 


(3) 


DETERMINATION    OF    ELECTROMOTIVE   FORCES         269 

Here,  again,  the  electromotive  force  of  C  must  be  greater 
than  of  either,  of  the  elements  A  or  B,  and  in  case  E1  is 
greater  than  E,  the  dispositions  of  these  two  elements  must  be 
interchanged.  The  sliding  resistance  EF  is  generally  made 
of  platinum-iridium  wire  of  constant  cross-section,  and  it  is  so 
arranged  that  it  can  be  minutely  subdivided.  That  employed 
by  Latimer-Clark  in  his  investigations  was  of  40  ohms  resist- 
ance, and  was  so  wound  on  an  insulated  cylinder  that  it  could 
be  read  to  one  twenty-thousandth  of  its  length. 

II.    Absolute  determination  of  electromotive  forces. 

From  the  relation  C=—  >  it  is  evident  that  if  C  and  R  be 
J\. 

expressed  in  absolute  units,  a  value  for  E  can  at  once  be 
deduced  in  the  same  system. 

So  far  as  C  is  concerned,  this  can  be  readily  accomplished  by 
the  use  of  a  tangent  galvanometer,  but  in  the  case  of  R,  since 
it  includes  the  internal  resistance  of  the  cell,  difficulties  arise  in 
connection  with  direct  determinations  which  make  the  results 
so  obtained  practically  worthless.  Hence,  methods  are  adopted 
in  which  a  determination  of  the  internal  resistance  is  avoided. 

For  constant  cells  Ohm's  method  may  be  followed,  but  it  is 
better  both  with  constant  and  variable  elements  to  apply  that 
outlined  by  Latimer-Clark,  which  is  a  method  of  compensation, 
the  cell  being  balanced  against  a  difference  of  potential  main- 
tained by  an  auxiliary  generator. 

(i)   OJims  method. 

The  circuit  consists  of  a  rheostat,  or  resistance  box,  a  tangent 
galvanometer,  and  the  element  whose  electromotive  force  is 
to  be  determined. 

A  convenient  deflection  B1  is  given  to  the  galvanometer  by 
suitably  adjusting  the  rheostat,  and  then  a  known  resistance 
R  is  added  to  the  circuit,  and  a  second  deflection  #2  is 
obtained. 


270 


EXPERIMENTAL   PHYSICS. 


Denoting  the  resistance  of  the  circuit  corresponding  to  the 
first  deflection  by  p,  it  follows  that 


and 


P+R 
K  being  the  constant  of  the  galvanometer. 

Eliminating  p  from  these  equations,  the  relation 


(i) 

(2) 


tan  0j  —  tan  02 
is  obtained,  and  from  it  E  can  be  found  in  absolute  units. 

(2)    Clark's  method. 

The  disposition  adopted  by  Latimer-Clark  is  similar  to  that 
devised  by  Poggendorff  for  comparative  measurements.  A  tan- 
gent galvanometer  G  (Fig.  49),  whose  constant  K  is  known, 


Fig.  49. 

is  inserted  along  with  a  rheostat  H  in  the  circuit  of  a  constant 
battery  B,  and  the  cell  A,  whose  electromotive  force  is  to  be 
found,  together  with  an  auxiliary  galvanometer,  form  the  derived 
circuit  CD. 


ABSOLUTE    DETERMINATION    OF   RESISTANCE.         271 

The  corresponding  terminals  of  the  two  cells  are  connected, 
so  that  they  both  tend  to  send  a  current  through  the  galva- 
nometer G  in  the  same  direction. 

The  rheostat  is  varied  until  the  galvanometer  F  indicates 
that  the  electromotive  force  of  A  is  balanced  by  the  potential 
difference  between  the  points  C  and  D.  It  only  remains  then 
to  note  the  reading  on  the  tangent  galvanometer,  and  to  measure 
the  resistance  of  the  partial  circuit  CLD.  Denoting  this  resist- 
ance by  R,  the  galvanometer  deflection  by  0,  and  the  electro- 
motive force  of  A  by  E,  these  quantities  are  connected  by  the 
relation 

E=KR  tan0. 

As  K,  R,  and  6  can  be  readily  determined  with  precision, 
the  method  affords  a  means  of  making  very  accurate  absolute 
determinations. 

The  method  is  exceedingly  simple,  and  should  any  errors  arise 
from  a  falling  off  in  the  intensity  of  the  current  from  B,  these 
may  be  corrected  by  closing  the  circuit  only  for  short  intervals 
at  a  time  during  the  test. 

Since  the  constant  K  is  given  by 


K, 


2  mr 


it  is  evident  that  the  method  may  be  applied  to  finding  H,  the 
horizontal  component  of  the  earth's  magnetic  action,  when  the 
laboratory  is  provided  with  a  cell  whose  electromotive  force  is 
accurately  known. 


XVII.    ABSOLUTE   DETERMINATION    OF    RESISTANCE 
BY    USE   OF   A   CALORIMETER. 

When  an  electrical  current  is  established  in  a  conductor  it 
is  produced  and  maintained  by  an  expenditure  of  energy,  me- 
chanical, thermal,  or  chemical,  and  consequently  it  may  itself 


272 


EXPERIMENTAL   PHYSICS. 


be  considered  to  be  a  form  of  energy  which  is  distributed  along 
the  conductor,  and  which  reappears  in  the  form  of  motion, 
chemical  separation,  or  heat. 

If  the  conditions  to  which  the  conductor  is  subjected  are 
such  that  the  transformation  cannot  take  place  into  the  first 
two  of  these  forms,  the  electrical  energy  will  be  expended  in  heat- 
ing the  conductor,  and  the  amount 
of  heat  so  produced  will  be  an 
exact  equivalent  of  the  work  done 
in  producing  the  current. 

The  conductor  may  then  be 
viewed  as  a  series  of  points  be- 
tween which,  while  a  current  is 
passing,  a  certain  amount  of  en- 
ergy is  being  expended,  and  the 
potential  difference  between  any 
two  points  will  represent  the  work 
done  between  them  in  a  unit  of 
time  when  the  electric  current  is 
of  unit  intensity. 
If,  then,  YI  and  V^  represent  in  volts  the  potentials  of  the 
two  terminals  of  a  conductor  forming  part  of  a  circuit,  and  C 
denotes  in  amperes  the  intensity  of  the  current  passing  between 
them,  C(V^  —  V^T  watts  will  represent  the  energy  transformed 
into  heat  in  the  conductor  in  T  seconds. 

Denoting  by  W  the  amount  of  this  energy  in  ergs,  we  have, 
W=C(Vl—  V^)T  io7  ergs,  and  combining  this  again  with  the 


• 

F 

1 

K 

>  ~  — 

El 
~£ 

S 

& 

# 

y 

_-3 

1? 

Fig.  50. 


relation  C- 


',  where  R  is  the  resistance  of  the  conductor 


in  ohms,  we  have, 

W=C*RT>  io7  ergs. 

The  mechanical  equivalent  of  heat  which  is  the  number  of 
units  of  work  required  to  be  expended  to  heat  one  gram  of 
water  through  one  degree  centigrade  has  been  found,  by  experi- 
ment, to  be  about  4.16  x  io7  ergs. 


ABSOLUTE    DETERMINATION   OF   RESISTANCE. 


273 


Adopting  this  value,  the  number  of  calories  or  units  of  heat 
developed  in  the  conductor  is  given,  therefore,  by 

calories, 


4.16-  10' 
or  H=C2RTx  •  24  calories, 

and  the  resistance  of  the  conductor  may  be  found  in  terms  of 
//,  T,  and  C  from  the  relation 


The  amount  of  heat  developed  in  the  conductor  is  ascertained 
by  immersing  it  in  a  known  mass  of  water,  and  by  noting  the 
gradual  rise  in  the  temperature  of  the  latter  when  the  current 
is  passed  through  it.  The  apparatus  is  similar  to  that  shewn  in 
Fig.  50.  The  water  is  contained  in  the  vessel  AB,  which  is 
made  of  sheet  brass,  and  in  order  to  reduce  the  errors  due 
to  radiation  and  absorption,  this  is  itself  placed  on  wooden 
supports  in  an  outer  vessel  CD,  made  of  the  same  material. 
The  conductor,  which  is  coiled  into  a  spiral,  is  attached  to  two 
insulated  binding  poles  inserted  in  the  cover  of  the  calorimeter, 
and  is  so  arranged  that  when  the  cover  is  in  position  it  is  com- 
pletely immersed  in  the  water,  and  yet  does  not  come  in  contact 
with  the  sides  or  bottom  of  AB.  The  apparatus  is  also  pro- 
vided with  a  delicate  thermometer  E  and  a  stirrer  F. 

In  conducting  the  experiment  the  weight  of  the  stirrer  and 
the  vessel  AB  is  first  determined  in  grams,  and  then  a  known 
mass  of  water  is  placed  in  it,  and  the  whole  lowered  into  the 
outer  vessel  CD.  The  cover,  with  coil  attached,  is  then  placed 
in  position,  and  the  thermometer  is  inserted  in  the  opening 
prepared  for  it. 

The  current  which  is  passed  through  the  conductor  may  be 
generated  by  a  powerful  battery,  or,  better  still,  by  a  dynamo, 
and  the  circuit  should  contain  either  a  sensitive  galvanometer, 


274  EXPERIMENTAL   PHYSICS. 

or  a  voltameter  and  a  galvanometer,  for  determining  its  inten- 
sity, as  well  as  a  rheostat  for  modifying  it  if  desired. 

When  the  initial  temperature  /1°  C.  of  the  water  has  been 
accurately  ascertained,  the  current  is  then  turned  on,  and  at 
the  same  instant  the  indication  on  a  stop  watch,  or  other  deli- 
cate time  recorder,  is  noted. 

By  means  of  the  rheostat  the  current  is  kept  constant,  and  at 
intervals,  as  the  experiment  goes  on,  both  the  time  and  the 
corresponding  temperature  of  the  water  is  noted,  care  being 
taken  to  keep  the  latter  well  stirred. 

The  temperature  of  the  water  should  always  be  initially  below 
that  of  the  room,  and  the  current  should  be  allowed  to  pass 
until  its  temperature  has  risen  as  much  above  that  of  the  room 
as  it  was  at  first  below  it. 

Denoting  the  final  temperature  by  /2°  C.  the  time  occupied  by 
the  experiment  by  T,  and  the  specific  heats  of  the  vessel  AB, 
and  the  conductor  by  sl  and  s%,  we  have,  if  m2  grams  is  the  mass 
of  the  conductor,  and  m1  and  m  those  of  the  vessel,  AB  and 
the  water  in  it  respectively, 

Jf=(M  +  mlsl  +  MySi)((z-tj,  (i) 

and  if  the  intensity  of  the  current  is  denoted  by  C,  the  resist- 
ance of  the  conductor  may  therefore  be  found  from  the  relation 

K—  4.10 


The  uncertainty  which  prevails  as  to  the  exact  value  of  the 
mechanical  equivalent  of  heat  rather  unfits  the  method  for  very 
accurate  determinations,  and  as  besides  the  resistance  of  a  con- 
ductor varies  with  its  temperature,  the  values  obtained  can  only 
be  considered  approximations.  The  method  is,  however,  instruc- 
tive, and  involves  an  excellent  training  in  the  use  of  the  funda- 
mental units  of  electrical  measurements. 

A  slight  modification  of  the  experiment  is  to  determine  the 
resistance  offered  by  the  carbon  filament  of  an  incandescent 


ABSOLUTE   DETERMINATION    OF    RESISTANCE. 


275 


lamp  to  the  passage  of  a  current.  This  may  be  accomplished 
by  covering  the  socket  of  the  lamp  with  paraffine  wax,  and 
noting  the  heat  evolved  when  it  is  immersed  in  the  water  in 
the  calorimeter.  The  results  obtained  illustrate  very  clearly 
that  the  resistance  of  a  carbon  conductor  rapidly  decreases  as 
its  temperature  increases. 


APPENDIX  A. 

DETERMINATION  OF  GRAVITY  BY  THE  PENDULUM. 

1.  If  a  pendulum  of  any  form  be  allowed  to  make  small  oscillations 
under  the  action  of  gravity,  we  have  the  time  of  a  complete  oscillation 

given  by  the  relation  t  =  2  TTA/— ,  where  /  is  the  length  of  the  equiva- 
*  S 

T  g     i      t2 

lent  simple  pendulum  and  equal  to  —     —     If,  now,  /  be  observed  by 

means  of  a  clock,  and  h  and  k  found,  we  have  the  value  of  g.  This 
method  is  one  of  the  most  accurate  known  for  finding  the  intensity  of 
the  earth's  attraction  at  different  points  on  its  surface.  Various  forms 
have  been  given  to  these  pendulums  from  time  to  time,  in  order  to 
insure  accuracy  of  measurement,  and  the  most  important  of  those 
which  have  been  used  for  the  scientific  determination  of  gravity  are 
described  below  : 

(a)    Borders  pendulum. 

Borda  (1792)  constructed  his  pendulum  so  as  to  realize  as  nearly  as 
possible  the  simple  pendulum.  It  was  made  of  a  sphere  of  known 
radius  (a).  To  render  it  very  heavy  it  was  composed  of  platinum,  and 
was  suspended  by  a  very  fine  wire  about  one  meter  in  length. 

The  knife  edge  which  carried  the  wire  and  sphere  was  so  arranged 
by  means  of  a  movable  screw  as  to  oscillate  in  the  same  time  as  the 
complete  pendulum.  The  time  was  determined  by  the  method  of 
coincidences,  and  g  was  found  from  the  relation 


g 

where  /  is  the  length  from  the  knife  edge  to  the  center  of  the  sphere, 
a  the  radius  of  the  sphere,  and  a  half  the  angle  of  a  single  oscillation. 

277 


278  EXPERIMENTAL   PHYSICS. 

(b)    Rater's  pendulum. 

In  1818,  Captain  Kater  determined  the  value  of  gravity  at  London 
by  applying  to  the  pendulum  the  principle  discovered  by  Huyghens, 
that  the  centers  of  suspension  and  of  oscillation  are  reversible. 

He  made  a  pendulum  of  a  bar  of  brass  about  an  inch  and  a  half 
wide  and  an  eighth  of  an  inch  in  thickness.  This  bar  was  pierced  in 
two  places,  and  triangular  knife  edges  of  hard  steel  were  inserted,  so 
that  the  distance  between  them  was  nearly  39  inches. 

A  large  mass  in  the  form  of  a  cylinder  was  placed  near  one  of  the 
knife  edges,  being  slid  on  by  means  of  a  rectangular  opening  cut  in  it  ;  a 
smaller  mass  was  also  attached  to  the  pendulum  in  such  a  way  as  to 
admit  of  small  motions  either  way.  The  pendulum  was  then  swung 
about  the  two  axes,  and  adjustment  of  the  masses  made  until  the 
time  of  small  oscillations  was  the  same.  This  time  being  noted, 
and  the  distance  between  the  knife  edges  being  accurately  measured, 
g  was  readily  calculated.  A  small  difference  being  generally  found  in 
the  two  times,  it  can  be  shewn  that  the  length  of  the  seconds  pendu- 
lum is  found  from  the  expression, 

(h, 


where  hlt  ht  are  the  distances  of  the  center  of  gravity  from  the  two 
knife  edges,  and  /„  t»  the  corresponding  times  of  oscillation. 

(c)    Repsold's  pendulum. 

It  was  noticed  in  experimenting  with  pendulums  made  like  Kater's, 
that  the  vibration  is  differently  affected  by  the  surrounding  air  according 
as  the  large  mass  is  above  or  below.  This  led  to  the  form  known  as 
Repsold's,  in  which  the  two  ends  are  exactly  similar  externally,  but 
the  pendulum  (which  is  cylindrical)  is  hollow  at  one  end.  The  center 
of  gravity  of  the  figure  is  equidistant  from  the  knife  edges,  but  the 
true  center  of  gravity  of  the  whole  mass  is  at  a  different  point. 

2.  Many  observers  have,  during  the  present  century,  conducted 
observations  at  different  points  on  the  earth's  surface  in  order  to  deter- 
mine not  only  the  length  of  the  seconds  pendulum  but  also  the 
excentricity  of  the  earth  considered  as  a  spheroid. 


APPENDIX   A. 


279 


Helmert  in  his  Geodesy  has  collected  the  results  of  nearly  all  the 
more  important  expeditions,  and  the  following  table  gives  some  of 
the  principal  stations,  with  the  corresponding  lengths  of  the  seconds 
pendulum  and  the  name  of  the  observer.  To  find  g  from  this  table 
for  any  place,  the  relation 

log  g  =  2  log  TT  +  log  / 

may  be  used,  where  /  is  the  length  of  the  seconds  pendulum  in  centi- 
meters. The  places  are  arranged  geographically  in  order  of  their 
latitudes,  and  shew  thereby  the  gradual  increase  in  the  length  of  the 
seconds  pendulum  as  we  go  from  the  Equator  to  the  Pole. 


PLACE. 

LATITUDE. 

/. 

OBSERVER. 

Rawak  
St.  Thomas  

o°     I'S. 
o     24  N. 
o     32  N. 
i     27  S. 
7     55  S. 
8     29  N. 
10     38  N. 
12    46  N. 
13      4N. 
15     56  S. 
17     56  N. 

22       32    N. 

22    55  S. 
33      2  S. 
34     54  S. 
38     28  N. 
40    44  N. 
41     41  N. 
43       7  N. 
44     50  N. 
45     24  N. 
48    40  N. 
5°     37  N. 
51     28  N. 
51     28  N. 
51     31  N. 
52     30  N. 
54    46  S. 
55     5i  S. 
55     58  N. 
57      3N. 
59    46  N. 
59    56  N. 
60    45  N. 

99.0966 
99.1134 
99.1019 
99.0948 
99.1217 
99.1104 
99.1091 
99.1227 
99.1168 
99.1581 
99.1497 
99.1712 
99.1712 
99.2500 
99.2641 

99-3°97 
99.3191 
99.3190 
99.3402 

99-347° 
99.3623 
99.3858 
99.4042 
99.4169 

99-4I43 
99.4140 

994235 
99.4501 

99-4565 
99-455° 
99.4621 
99.4854 
99.4876 
99-4959 

Freycinet. 
Sahine. 
Hall. 
Foster. 

Sabine. 

Basevi  and  Heaviside. 
Basevi  and  Heaviside. 

Sabine. 
Basevi  and  Heaviside. 

Liitke. 
Foster. 
Biot. 

Duperrey. 
Biot. 
Biot. 

Kater. 

Foster. 
Foster. 

Liitke. 
Sawitsch. 

Galapagos  
Para  

Trinidad  
Aden  
Madras  
St.  Helena  
Jamaica  
Calcutta  
Rio  Janeiro  

Valparaiso  

Lipari  
Hohoken  N.  J  

Tiflis  
Toulon  
Bordeaux  
Padua  

Paris  
Shanklin  Farm  (Isle  of  Wight)  . 
Kew  ... 

Greenwich  
London  
Berlin  
Staten  Island  
Cape  Horn  
Leith 

Sitka  .  ... 

Pulkovva  
Petersburg  
Unst  

28o  EXPERIMENTAL    PHYSICS. 

Those  places  in  the  preceding  table  for  which  the  lengths  of  the 
seconds  pendulum  have  been  calculated  from  a  number  of  observations 
made  by  different  observers  are  indicated  by  a  dash. 

3.  During  the  past  few  years  several  observers  have  made  observa- 
tions on  the  value  of  g  at  different  points  in  North  America.     Professor 
Mendenhall,  of  the  U.  S.  Coast  Survey,  during  the  summer  of  1891, 
visited  a  number  of  places  on  the  Pacific  coast  between  San  Francisco 
and  Alaska,  and  in  his  report  of  the  expedition  gives  a  table  of  the 
values  determined,  with  the  places  and  corresponding   latitudes.     He 
made  use  of  a  half-seconds  pendulum,  inclosed  in  an  air-tight  chamber, 
which  could  be  exhausted  with  an  air-pump.     A  special  method  was 
used  for  noting  the  coincidences.     (See  U.  S.  Coast  and  Geodetic  Survey, 
Report  for  1891,  part  2.) 

4.  Defforges,  one  of  the  greatest  living  authorities  on  methods   of 
gravity  determination,  crossed  from  Washington  to  San  Francisco  during 
the  summer  of  1893,  and  made  a  number  of  observations,  which  are 
given  in  the  following  table.     The  value  of  g  alone  is  given  : 

Washington 980.169 

Montreal 980.747 

Chicago 980.375 

Denver 980.983 

Salt  Lake  City 980.050 

Mt.  Hamilton 979. 916 

San  Francisco 980.037 

These  are  all  reduced  to  sea-level. 

NOTE.  —  An  excellent  book  of  reference  on  the  subject  of  gravity  determination 
is  Memoir es  rtlatifs  h  la  Physique,  Tome  IV,  Paris,  1889. 


APPENDIX    B. 

THE   TORSION   PENDULUM. 

IN  many  instruments  of  precision  designed  for  investigations  in  Elec- 
tricity, Magnetism,  and  Gravitation,  certain  portions  of  the  apparatus, 
which  are  movable  about  a  vertical  axis,  are  suspended  either  by 
metallic  wires  or  elastic  fibers. 

The  suspension  may  be  either  unifilar  or  bifilar.  If  the  former  method 
is  adopted  and  suspension  is  all  that  is  required,  fibers  are  selected,  such 
as  silk  cocoon  threads,  which  offer  practically  no  resistance  to  torsion, 
and  are  capable  of  supporting  a  considerable  mass.  If,  however,  a  couple, 
or  a  number  of  couples,  is  applied  to  the  movable  part  of  the  instrument, 
and  this  system  is  equilibrated  and  measured  by  the  torsion  in  the  sus- 
pending fibers  or  wires,  the  latter  are  generally  made  from  some  metal, 
such  as  silver,  or  from  fine  glass  or  quartz  threads. 

In  the  bifilar  suspension  the  movable  apparatus  is  attached  to  two 
threads  or  fibers  of  the  same  length,  which  have  their  upper  ends  fas- 
tened to  two  fixed  points.  When  the  suspended  system  is  displaced 
from  its  position  of  rest,  it  is  its  weight  which  gives  rise  to  the  direc- 
tive force,  and  the  restorative  couple  is  but  slightly  affected  by  any 
torsion  which  may  exist  in  the  suspending  threads. 

The  elastic  properties  of  substances  used  for  suspension  purposes  may 
be  investigated  by  means  of  the  Torsion  Pendulum.  In  such  an  appara- 
tus a  wire  or  fiber  of  the  material  in  question  is  suspended  from  some 
point  to  which  it  is  firmly  attached,  and  a  heavy  body  called  a  vibrator 
is  securely  fastened  to  its  lower  extremity. 

Various  forms  have  been  suggested  for  this  vibrator.  It  is  generally 
made  of  some  non-magnetic  substance  such  as  brass  or  copper,  and  it 
must  be  of  such  dimensions  that  its  moment  of  inertia  about  the  axis  of 
the  suspending  wire  can  be  readily  calculated.  Professor  Andrew  Gray 
recommends  a  hollow  circular  cylinder  of  brass  or  copper,  as  it  is  best 
suited  for  a  correct  determination  of  the  moment  of  inertia,  and  as 
besides  its  oscillations  are  but  slightly  affected  by  the  presence  of  the  air. 

281 


282  EXPERIMENTAL   PHYSICS. 

Professor  Threlfall,  in  his  article*  on  "  The  Elastic  Constants  of  Quartz 
Threads,"  states  that  the  vibrators  used  by  him  in  his  investigations 
"were  (i)  a  pure  silver  anchor  ring  supported  by  a  disc  of  aluminium 
foil  through  the  center  of  which  passed  a  short  bit  of  aluminium  wire 
to  which  the  thread  was  attached,  (2)  a  cylindrical  vibrator  of  brass, 
vibration  being  about  the  axis  of  the  cylinder,  and  (3)  a  disc  of  brass, 
vibration  taking  place  about  its  axis  of  revolution." 

When  the  vibrator  is  given  a  rotation  about  the  axis  of  the  suspending 
wire  or  thread,  and  is  then  left  to  itself,  it  will  perform  oscillations  about 
its  mean  position  which  within  the  limits  of  perfect  elasticity  are  isochro- 
nous. This  at  once  indicates  that  the  motion  is  harmonic,  and  that 
therefore  the  moment  of  the  restorative  couple  is  proportional  to  the 
angle  through  which  the  suspended  body  is  rotated.  This  moment  is 
also  proportional  to  the  fourth  power  of  the  diameter  of  the  wire,  and 
varies  inversely  as  its  length.  As  formulated  originally  by  Coulomb 
these  laws  are  exhibited  in  the  relation 


where  d  is  the  diameter  of  the  wire,  /  its  length,  and  0  the  angular  dis- 
placement. 

The  time  of  an  oscillation  is  therefore  given  by 

-5V? 

K  being  the  moment  of  inertia  of  the  vibrator  about  the  axis  of  rotation. 

In  order  to  verify  these  laws,  it  is  only  necessary  to  select  a  number 
of  wires  of  different  dimensions,  but  of  the  same  material.  If  the  vibra- 
tor be  suspended  by  each  of  them  in  turn,  and  the  time  of  an  oscillation 
observed  with  each,  it  will  be  found  that  if  these  values  together  with 
the  corresponding  ones  for  d,  /,  and  K  be  substituted  in  equation  (2),  a 
constant  value  will  be  obtained  for  p.. 

This  quantity,  which  is  termed  the  constant,  or  coefficient  of  torsion, 
depends  only  on  the  nature  of  the  suspending  wire,  and  on  its  tempera- 
ture. In  the  C.G.S.  system  of  units  it  is  the  numerical  expression  for 
the  couple  which,  if  applied  to  one  base  of  a  circular  cylinder  of  wire, 
whose  length  and  diameter  are  each  one  centimeter,  would  twist  this 
base  through  a  unit  angle,  the  other  base  being  fixed. 

*  Phil.  Mag.,  July,  1890. 


APPENDIX    B. 


283 


The  following  table  gives  the  values  of  this  coefficient  for  a  number  of 
metals,  and  for  silk  and  quartz  fibers  : 


SUBSTANCE. 

COEFFICIENT  OF  TORSION. 

SUBSTANCE. 

COEFFICIENT  OF  TORSION. 

Aluminium 
Silver 

2.5516  x  io10 

2.6144 

Copper 
German  Silver 

4.2458  x  io10 

4.7412 

Gold 

2.6958 

Platinum 

6.6659 

Zinc 

3-2559 

Iron 

7.4448 

Brass 

3-3727 

Quartz  (fibers) 

2.8289 

Platinum-Silver 

(lPt  +  2Ag) 

3.5581 

*  Silk  (fibers) 

*  Professor  T.  Gray  in  1886  made  a  thorough  investigation  of  the  elastic  properties 
of  silk  threads,  and  found  that  the  torsion  factor  for  fibers  of  Japanese  silk  one  centi- 
meter in  length,  and  of  diameters  ranging  from  .0008  to  .0015  cms.  varied  from  .00096 
to  .0025  of  a  dyne. 


Metallic  wires  have  been  found  quite  unsuitable  for  suspension  pur- 
poses when  delicate  manipulation  is  demanded.  Their  coefficients  of 
torsion  vary  considerably  with  changes  in  temperature,  and  are  slightly 
diminished  when  the  wires  have  been  allowed  to  oscillate  for  a  long 
period.  Permanent  deformations  also  soon  appear  in  them  unless  the 
amplitudes  of  the  oscillations  are  very  small,  and  consequently  the 
angular  displacements  cannot  be  relied  on  in  making  accurate  measure- 
ments of  applied  couples. 

The  latter  difficulty  was  experienced  by  Professor  C.  V.  Boys*  in  his 
recent  investigation  on  the  Newtonian  Constant  of  Gravitation.  After 
experimenting  with  many  substances  he  found  fibers  of  quartz  to  possess 
elastic  properties  more  nearly  perfect  than  those  of  any  other  material. 
Such  fibers  can  be  drawn  to  any  required  length,  and  so  exceedingly  fine 
that  even  a  powerful  microscope  fails  to  reveal  their  presence.  They 
are,  however,  very  regular,  are  as  strong  as  steel,  and  remain  constant  for 
a  long  period.  With  them  the  suspended  body  can  be  permitted  to 
oscillate  through  large  angles  without  any  deformation  or  displacement 
of  the  zero  position  appearing.  On  account  of  these  many  excellent 
properties,  they  are,  therefore,  most  suitable  for  suspensions  in  investiga- 
tions demanding  accuracy  and  delicacy  of  manipulation. 

When  a  magnet  whose  magnetic  moment  is  known  is  suspended  in  a 
field  of  determined  intensity,  the  coefficient  of  torsion  can  also  be  found 


*  Phil.  Mag.,  June, 


284  EXPERIMENTAL   PHYSICS. 

by  observing  the  deflections  of  the  magnet  from  its  position  of  rest  when 
the  suspending  fiber  or  wire  is  subjected  to  different  amounts  of  torsion. 
If  0  denote  the  deflection  of  the  magnet  corresponding  to  the  torsion 
in  the  wire  <£,  and  H  and  M  the  intensity  of  the  field  and  the  magnetic 
moment  respectively,  these  quantities  are  connected  by  the  relation 


(3) 


and  the  constant  of  torsion  is  therefore  given  by 
HML  sin  0 


When  this  method  is  adopted,  care  must  be  taken  to  have  the  magnet 
initially  at  rest,  without  the  suspending  wire  being  subjected  to  any 
torsional  strain.  Further  details  of  the  method  have  already  been  given 
on  page  187. 

Determination  of  moments  of  inertia. 

By  a  reference  to  equation  (2)  it  can  be  seen  that  if  the  constant  of 
torsion  for  a  given  wire  /*.  is  known,  the  moment  of  inertia  K  of  the  sus- 
pended body  can  be  at  once  deduced  by  finding  the  time  of  an  oscilla- 
tion. 

It  is  better,  however,  owing  to  the  difficulty  in  obtaining  a  value  for  p 
sufficiently  exact  for  this  purpose,  to  adopt  a  comparative  method.  The 
body  whose  moment  of  inertia  is  required  is  first  attached  to  the  sus- 
pending wire,  and  its  time  of  oscillation  found.  It  is  then  replaced  by 
some  body  whose  moment  of  inertia  is  known  or  can  be  readily  calcu- 
lated, and  the  oscillation  period  is  again  determined. 

Since  the  directive  couple  in  the  two  cases  is  the.  same,  the  constant 
of  torsion  can  then  be  eliminated.  If  /,  and  /  are  the  two  periods  of 
oscillation,  and  Kl  and  K  the  required  moment  of  inertia  and  that  which 
is  known  respectively,  it  follows  from  equation  (2)  that 

K,  =  K'±. 

A  modification  of  this  method  is  to  use  a  bar  magnet  for  the  auxiliary 
body,  and  to  determine  the  vibration  period  first  with  the  given  body 
attached,  and  then  with  it  removed.  The  restorative  couple  will  in  this 
case  depend  on  the  intensity  of  the  magnetic  field  as  well  as  on  the 
coefficient  of  torsion  of  the  wire.  The  vibrations  must  also  be  taken  of 
small  amplitude. 


TABLES. 


TABLE   I. 

5,  £=.3183, 
77^  =  9.8696,   VTT  =1.7724. 
Circle,  diameter  2  a : 

circumference  =  2  ira, 
area  =  77-0*. 
Ellipse,  axes  2  a,  2  b : 

area  =  irab. 
Triangle,  base  b,  altitude  a  : 

area  =  \  ab. 
Sphere,  radius  a : 

surface  =  4  ira2, 
volume 


Right  circular  cylinder,  radius  a,  length  I : 

surface  (including  ends)  =  2  ira(l+  a), 
volume  =  7T02/. 

Right  circular  cone,  radius  of  base  b,  altitude  a  : 

surface  =  TT£(  V<z2-M2+  b), 
volume  =     -nab*. 


TABLE    II. 

i  meter       =  100  centimeters  =  1000  millimeters. 

i  liter          =  1000  cubic  centimeters. 

i  kilogram  =  1000  grams. 

i  gram        =  10  decigrams  =  100  centigrams  =  1000  milligrams. 

285 


286 


EXPERIMENTAL   PHYSICS. 

i  meter  (i  m.)  =39-37*  inches, 

i  inch  =  25.4  millimeters, 

i  millimeter  (i  mm.)  =.039371  inches. 

i  cubic  centimeter  of  water  at  4°C.  =  i  gram. 

i  cubic  inch  =  16.386  cubic  centimeters  (cc.). 
i  pint  =  567.93  cubic  centimeters, 

i  liter  =1.76  pints. 

i  ounce  avoirdupois  =  28.35  grams- 
i  pound  avoirdupois  =  453.593  grams. 
i  gram  =  15.432  grains, 

i  kilogram  =  2.2046  pounds. 


TABLE    III. 


Specific   Gravities  of  Solids. 


Amber     
Antimony     .... 
Bismuth  

i.i 

.     6.7 
.     o  8 

Iceland  spar       .... 
India  rubber      .... 
Iron  (cast)    .                    -. 

2-7 

•99 

7.2 

Bone  .                ... 

I  0 

Iron  (wrought) 

7-70 

Brass     | 

84 

Iron  (steel)        .... 

7-79 

Bronze  j 
Carbon  (gas) 

i  8 

Ivory   
Lead 

1.92 

114 

Copper    . 

.     8.QS 

Lignum  vitse       .... 

I.* 

Cork  
Diamond      .... 
Ebony      
German  silver 

.24 

•     3-5 
.     1.19 
8  62 

Nickel      
Platinum        
Sand    
Silver  (925  fine) 

8-57 
21.5 
1.42 
10.18 

Glass  (°reen) 

2  64. 

Silver  (pure) 

IO.^  7 

Glass  (crown) 

2   C 

Starch 

I  .cr  -2 

Glass  (flint)      .     .     . 
Gold  (18  caret) 

•     3-3 
14  88 

Sugar  (cane)      .... 
Salt 

1.6 

.Q2 

Gold  (pure) 

Tin 

7.3 

Graphite       .... 
Gunpowder 

2.2 
2.O? 

Wax  (bees')       .... 
Zinc  (rolled)      .... 

.96 

7.2 

Guttapercha 

Zinc  (cast) 

6.86 

Human  body    . 

I.O7 

TABLES. 


28; 


Specific   Gravities  of  Liquids. 

Alcohol  (ethyl)  at  15°  C 79 

Ammonia  at  15°  C 761 

Benzine  at  15°  C 89 

Bisulphide  of  carbon  at  15°  C.       .     .     .     1.28 

Chloroform  at  15°  C 1.5 

Ether  at  15°  C 72 

Glycerine  at  15  C 1.26 

Hydrochloric  acid  ati5°C 1.2 

Mercury  at  zero  C.      .     .     .     .     .     .     .   13.596 

Milk  (cows')  at  zero  C 1.03 

Nitric  acid  at  15°  C 1.52 

Olive  oil  at  15°  C 915 

Sea  water  at  o°  C 1.026 

Turpentine  at  15°  C 872 

Water  at  4°  C i. 

Specific  Gravities  of  Gases  referred  to  Air  at  o°   C.  and  760  mm. 

Air i. 

Oxygen 1.10563 

Nitrogen 97J37 

Hydrogen 06926 

Carbon  dioxide 1.52901 

Vapours. 

Alcohol  (ethyl) 1.61  at    78°.4  C. 

Chloroform    ......     4.20  at    6o°.8 

Water 64  at  100° 

Ether   ........     2.59  at    35°.5 

Iodine 8.72  at  175° 

Mercury    .     .     „     .     .     .     .     6.98  at  350° 

i  liter  of  dry  air  at  o°  C.  and  760  mm.  pressure  weighs  1.293  grams. 


288 


EXPERIMENTAL    PHYSICS. 


TABLE   IV. 

Specific  Heats  of  Solids  and  Liquids  referred  to   Water. 
Aluminium  .202     i 


Liquids  at  15°   C. 

Alcohol.     .    ..     .     ...       .615 

Chloroform 233 

Ether     ...".• 517 

Mercury 033 

Gases  at  Constant  Pressure. 

Air    .     .     ...     .    V    .     .     .2375 

Steam 4805 


Antimony  
Bismuth  
Brass 

.     .0507 
•     -0305 

.OQ4 

Copper  

Glass 

.     .095 

2 

Gold 

O"?24 

Graphite 

Ice  
Iron  ...  ... 

•     -5 

.112 

Lead  .  . 

.03  I  tJ 

Platinum  
Quartz 

•       .0356 

Silver  
Sulphur  
Tin  . 
Zinc. 

•       -0559 
.       .184 
•       -0559 

.GO  ^=; 

TABLE   V. 
Latent  Heats  of  Fusion. 

Beeswax 97.22 

Lead 5.37 

Steam 536. 

Sulphur 9.35 

Water 79-25 

Zinc 28.15 

TABLE   VI. 

Cubical  Expansion  of  Solids  and  Liquids  for  i°  C. 

Copper 00005 

Glass 000023—28 

Iron 0000355 

Platinum 000026 

Silver 0000583 

Tin 000069 

Zinc     .  .000089 


TABLES.  289 


Liquids  (Mean  Expansion). 

Alcohol 00108 

Chloroform 0014 

Ether 0021 

Mercury      .     .     ...     .0001815 

Turpentine 00105 

Water 00012  at  15°  C.  to 

.00074  at  90°  C. 

Gases  expand  .003665  of  their  volume  for  each  degree  Centigrade. 


TABLE  VII. 

Expansion  of  Water  from  o°  to  100°. 
Volume  of  i    Gram  of  Water  in   Cubic  Centimeters. 

TEMPERATURE.  VOLUME.  INCREASE  PER  i". 

O°  I.OOOI 

4      ....     i.oooo 
10  ...     i.ooo-? 

J        .....  00012 
1C         ....       I.OOO9  , 

y    .....  00016 

20      •     •     •     •     I'°OI  .     .00024 

.  .     .00028 

3°      •     •     •     •     I'°°43     ,  ,     .00032 

35       '     '     '     '     I'°°59  .00036 

40         ....       1.0077 

'  '        .....  00040 

4?       ....     1.0097 

,     .     .     .00046 

SO         ....       I.OI20 

.....  00048 
CC         .       .       .       .       I.OI44 

.....  OOOC2 
60         ....       I.OI70 

.....  OOOS4 
65         ....       I.OI97 

V/        .....  00060 
7O         ....       I.O227  ,. 

'       .....  00062 
715         ....       I.O2i;8  .. 

J          .....  00064 
80         ....        1.  0200 

9  .....  00066 


„     .....  00070 
oo       ....     1.0358 

OJ   .....  00074 
95   ....   1.0^9=5 

J7J   .....  00074 
100   ....  1.0432 


u 


2QO 


EXPERIMENTAL   PHYSICS. 


TABLE   VIII. 

Boiling  Temperature  of  Water  (/)   at  Barometer  Pressure 
(after  Regnaulf). 


b 

t 

b 

/ 

b 

/ 

b 

/ 

b 

t 

680 

96°92 

700 

970.72 

720 

98°49 

740 

99°.26 

760 

IOO°.00 

68  1 

.96 

701 

•75 

721 

•53 

741 

.29 

761 

.04 

682 

97  .00 

702 

•79 

722 

•57 

742 

•33 

762 

.07 

683 

.04 

7°3 

•83 

723 

.61 

743 

•37 

763 

.11 

684 

.08 

704 

.87 

724 

•65 

744 

.41 

764 

•15 

685 

.12 

7°5 

.91 

725 

.69 

745 

•44 

765 

.18 

686 

.16 

706 

•95 

726 

.72 

746 

.48 

766 

.22 

687 

.20 

707 

•99 

727 

.76 

747 

•52 

767 

.26 

688 

.24 

708 

98  -03 

728 

.80 

748 

•56 

768 

•29 

689 

.28 

709 

.07 

729 

.84 

749 

•59 

769 

•33 

690 

•32 

710 

.11 

73° 

.88 

75° 

•63 

770 

•36 

691 

.36 

711 

•15 

73i 

.92 

75  ! 

.67 

771 

.40 

692 

.40 

712 

.19 

732 

•95 

752 

•7° 

772 

•44 

693 

•44 

7*3 

.22 

733 

•99 

753 

•74 

773 

•47 

694 

.48 

7H 

.26 

734 

99  -03 

754 

.78 

774 

•51 

695 

•52 

7i5 

•3° 

735 

.07 

755 

.82 

775 

•55 

696 

•56 

716 

•34 

736 

.11 

756 

•85 

776 

•58 

697 

.60 

717 

•38 

737 

.14 

757 

.89 

777 

.62 

698 

.64 

718 

.42 

738 

.18 

758 

•93 

778 

•65 

699 

.68 

719 

.46 

739 

.22 

759 

.96 

779 

.69 

700 

97°.72 

720 

98°49 

740 

99.°26 

760 

IOO°.OO 

780 

IOO°.72 

TABLES. 


29I 


TABLE    IX. 
For  Hygrometry. 

Pressure  of  aqueous  vapour  e  and  weight  of  water  f  contained  in  I  cubic  meter  of 
air  with  dew-point  /;  or,  when  at  the  temperature  /,  the  air  would  be  saturated  with 
aqueous  vapour. 


t 

• 

/ 

t 

' 

/ 

t 

• 

/ 

t 

' 

/ 

-10° 

mm. 
2.0 

2.1 

0° 

mm. 
4.6 

gr- 
4-9 

10° 

9-i 

gr- 
9-4 

20 

mm. 
17.4 

gr- 
I7.2 

-  9 

2.2 

2-4 

I 

4-9 

5-2 

ii 

9.8 

IO.O 

21 

I8.5 

18.2 

-  8 

2.4 

2.7 

2 

5-3 

5-6 

12 

10.4 

10.6 

22 

19.7 

19-3 

-  7    2.6 

3.O 

3 

5-7 

6.0 

13 

ii.  I 

"•3 

23 

20.9 

20.4 

-  6   2.8 

3-2 

4 

6.1 

6.4 

14 

11.9 

12.0 

24 

22.2 

21.5 

-5    3-i 

3-5 

5 

6.5 

6.8 

«5 

12.7 

12.8 

25 

23.6 

22.9 

-  4 

3-3 

3-8 

6 

7.0 

7-3 

16 

13-5 

13-6 

26 

25.0 

24.2 

-  3 

3-6 

4.1 

7 

7-5 

7-7 

17 

14.4 

14-5 

27 

26.5 

25.6 

-  2 

3-9 

4-4 

8 

8.0 

8.1 

18 

15.4 

I5-I 

28 

28.1 

27.0 

-  i 

4.2 

4.6 

9 

8-5 

8.8 

19 

16.3 

16.2 

29 

29.8 

28.6 

0° 

4.6 

4-9 

10 

9.1 

9-4 

20 

17.4 

17.2 

30 

31-6 

3O.I 

TABLE    X. 


SCALE  OF  PHYSICISTS. 


ut 

256 

512 

ut 

256 

re 

288 

576 

ut* 

271.22 

mi 

320 

640 

re 

287-35 

fa 

341-33 

682.66 

re* 

3°4-437 

sol 

384 

768 

mi 

322-539 

la 

426.66 

853-33 

fa 

34i-7J9 

si 

480 

960 

fa* 

362.038 

sol 

383-566 

sol* 

406.374 

la 

43°-539 

la* 

456.140 

si 

483.263 

SCALE  OF  EQUAL  TEMPERAMENT. 


292 


EXPERIMENTAL   PHYSICS. 


TABLE   XI. 

Mean  Indices  of  Refraction,  and  Dispersions  of  Several  Bodies. 


INDEX  OF  REFRACTION. 
•       •       1-53       •       • 


Crown  glass  (mean)  .... 

Flint  glass i  .60  .     . 

Water 1.336  .     . 

Alcohol r-372  •     • 

Carbon  disulphide 1.68  .. 

Canada  balsam 1.54  .     . 

Air 1.000294 . 


DISPERSION. 
.022 
.042 
.0132 


.0837 


TABLE  XII   A. 

Elements  of  Terrestrial  Magnetism  at  Toronto. 
Latitude,  43°  39'  36".     Longitude,  5  h.  17  m.  34.65  s. 


DATE. 

DECLINATION. 

INCLINATION. 

HORIZONTAL  FORCE 
IN  C.G.S.  UNITS. 

(westerly) 

I885 

3°S9'-8 

74°  5  *  '-6 

•I65579 

1886 

4°      2'.I 

74°  48'-9 

.165717 

1887 

4°    4'.» 

74°  47  '-6 

•165875 

1888 

4°    8'.3 

74°  46  '-5 

•165993 

1889 

4°  1  2  '.o 

74°  44  '-7 

.l66lll 

1890 

4°  i8'.2 

74°42(.2 

.166150 

1891 

4°  23'-3 

74°37'-5 

.166170 

1892 

4°  29'.2 

74°  37'-2 

.166229 

1893 

4°  36'-4 

74°  36'.2 

.166354 

1894 

4°  42  '-3 

74°  34  '-7 

.166334 

TABLES. 


293 


TABLE    XII   B. 
Elements  of  Terrestrial  Magnetism  at  Other  Places. 


DATE. 

PLACE. 

DECLINATION. 

INCLINATION. 

HORIZONTAL  FORCE 
IN  C.G.S.  UNITS. 

(westerly) 

1887 

Greenwich 

1  7°  49  '.i 

67°  26'  26" 

.18175 

1888 

« 

i7°4o'.4 

67°  25'  25" 

.18204 

l889 

" 

I7°34'-9 

67°  24'    9" 

.18201 

1890 

" 

i7°28'.6 

67°  22'  52" 

.18232 

1891 

" 

i7°a3'-4 

67°  2l'  24" 

.18254 

1889 

Washington 

4°i'3i" 

7i°    5'59" 

.198693 

1890 

a 

4°  5  '45" 

?i°    4'3lM 

.198604 

1891 

" 

4°  9'  43" 

7i°    5'    4" 

.198550 

1892 

<i 

4°  14'  12" 

7i°    3'55" 

.198485 

1886 

Paris 

1  6°  oo  '.9 

65°   i5'.« 

•J9439 

1887 

it 

i5°54'-8 

65°    i4'-7 

.19470 

1888 

u 

i5°49'-7 

65°    i4'-5 

.19496 

l889 

" 

15°  44  '.6 

65°     I2'.6 

.19522 

1890 

u 

i5°38'.7 

65°    u'. 

•I9543 

1891 

It 

i5°32'-8 

65°    IQ'.I 

•19S58 

294 


EXPERIMENTAL   PHYSICS. 


TABLE   XIII. 

Cross-Section   of  Round  Wires,  with    Resistance,  Conductivity,  and  Weight  of 
Pure  Copper  Wires,  according  to  the  Birmingham  Wire  Gauge. 

TEMPERATURE  15°  C. 


B. 
W.G. 

DIAMETER. 

AREA  OF 
CROSS-SECTION. 

RESISTANCE. 

CONDUC- 
TIVITY. 

WEIGHT 
(Density  =  8.95). 

Ins. 

Cms. 

Sq.  Ins. 

Sq.  Cms. 

Legal 
Ohms' 
per 
Yard. 

Legal 
Ohms 

MPe1e, 

Yards 
per 

te! 

Meters 
per 
Legal 
Ohm. 

Lbs. 
Y^rd. 

Grams 
Meter. 

0000 

•454 

I-I53 

.162 

1.0444 

.000150 

.000165 

6640 

6072 

1.884 

934-7 

000 

•425 

1.079 

.142 

.915 

.000172 

.000188 

5819 

5321 

1.651 

819-1 

oo 

.380 

•965 

•"3 

•732 

.000215 

.000235 

4653 

4254 

1.320 

654.8 

0 

•34° 

.864 

.0908 

.586 

.000269 

.000294 

3735 

3415 

1.056 

524.2 

j 

.300 

.762 

.0707 

•456 

.000345 

.000378 

2899 

2652 

.822 

408.1 

2 

.284 

.721 

.0633 

.409 

.000385 

.000420 

2599 

2377 

•737 

365-8 

3 

.259 

.658 

.0527 

•340 

.000463 

.000506 

2162 

1996 

•613 

304.2 

4 

5 

.238 

.220 

.605 
•559 

•0445 
.0380 

.287 
•245 

.000642 

.000599 
.000701 

1825 
1561 

1669 
1427 

.518 
•442 

256.9 
219-5 

6 

203 

.516 

.0324 

.209 

.000754 

.000824 

1328 

1214 

•377 

186.9 

7 

!i8o 

•457 

.0254 

.164 

.000958 

.00105 

1044 

1004 

.296 

146.9 

8 

.165 

.419 

.0214 

.138 

.00114 

.00125 

877 

802 

.249 

123.5 

9 

.148 

.376 

.0172 

.00155 

706 

645 

99-3 

•  134 

•34° 

.0141 

.0910 

.00173 

.00189 

578 

529 

S«J 

Eu4 

, 

.120 

•3°5 

.0113 

.0730 

.00216 

.00235 

463 

424 

•132 

65-5 

2 

.109 

.277 

•00933 

.0602 

.00261 

.00286 

382 

350 

.109 

53-9 

3 

•095 

.241 

.00709 

•0457 

.00344 

.00376 

291 

266 

.0825 

40.9 

4 

•083 

.211 

•00541 

•°349 

.00451 

.00492 

221 

203 

.0630 

31.2 

5 

.072 

.183 

.00407 

.0263 

.00599 

.00655 

l67 

J53 

.0474 

23-5 

6 

•065 

.165 

.0033! 

.0214 

•00735 

.00804 

I36 

124 

.0386 

19.2 

7 

.058 

•147 

.00264 

.0170 

.00923 

.0101 

108 

98.7 

.0307 

I5.3 

8 

.049 

.124 

.00189 

.0122 

.0130 

.0141 

77-3 

70.7 

.0220 

10.9 

9 

.042 

.107 

.00139 

.00894 

.0176 

.0194 

5  -8 

52.O 

.Ol6l 

•035 

.0889 

.000962 

.00621 

.0253 

.0277 

39-4 

36.1 

.0122 

5-Sfi 

t 

.032 

.0813 

.000804 

.00519 

.0304   ,  .0331 

32.0 

30.1 

.00936 

4-64 

2 

.028 

.0711 

.000616 

.00397 

•°395 

•0433 

25-3 

23-1 

.00716 

3-55 

3 

.025 

.0635 

.000491 

.00317 

.0496 

•0543 

20.2 

18.4 

.00571 

2.83 

4 

.022 

•0559 

.000380 

.00245 

.0642 

.0701 

15-6 

14-3 

.00442 

2.19 

5 

.020 

.0508 

.000314 

.00203 

.0778 

.0849 

12.8 

11.7 

.00367 

1.82 

26 

.0  8 

.0457 

.000254 

.00164 

•°959 

.105 

102 

9-53 

.00296 

1-47 

27 

.0  6 

.0406 

.000201 

.00130 

•133 

8^25 

7-54 

.00234 

1.16 

28 

.0  4 

.0356 

.000154 

.000993 

ilp 

•173 

6.3I 

5-77 

.00179 

.889 

29 

•o  3 

.0330 

.000133 

.000856 

.184 

.201 

541 

4.98 

.00154 

.766 

30 

.0  2 

•0305 

.000732 

.216 

•235 

4.64 

4.24 

.00132 

.653 

3i 

.010 

.0254 

.0000785 

.000507 

•3" 

•339 

3-23 

2.95 

.000915 

•454 

32 

.009 

.0229 

.0000636 

.000410 

•384 

.419 

2-51 

2.39 

.000746 

•367 

33 

.008 

.0203 

.0000503 

.000324 

.486 

•530 

2.06 

1.88 

.000585 

.290 

34 

.007 

.0178 

.0000385 

.000248 

•634 

.693 

1.58 

i-45 

.000442 

1 

.005 
.004 

.0127 

.0000196 

.000127 
.OOOoSlI 

1.25 
1.94 

'•35 
2.13 

.806 
.516 

•736 
•47' 

««2 

•"3 
.0726 

TABLES. 


295 


TABLE   XIV. 

Cross-Section  of  Round  Wires,  with  Resistance,  Conductivity,  and  Weight  of 
Hard-Drawn  Pure  Copper  Wires,  according  to  the  New  Standard  Wire  Gauge 
(Legalized  Aug.  23,  1883,  Great  Britain  and  Ireland). 

TEMPERATURE  15°  C. 


1 

DIAMETER. 

AREA  OF 
CROSS-SECTION. 

RESISTANCE. 

CONDUC- 
TIVITY. 

WEIGHT 
(Density  =  8.95). 

f 

Ins. 

Cms. 

Sq.  Ins. 

Sq.  Cms. 

Legal 
Ohms 

YPard. 

Legal 
Ohms 
per 
Meter. 

Yards 
per 
Legal 
Ohm. 

Meters 
per 
Legal 
Ohm. 

Lbs. 

Grams 
Meier. 

0000000 

.500 

I270  u-  > 

1.267 

.000125 

.000136!  8055 

7365 

.285 

"34 

000000 
00000 

.464 
•432 

I.'^f  '  15 

1.091 

.946 

.000144 
.ooo!66 

.0001571  6937 
.000182  6013 

6343 
S498 

.970 
.706 

976.3 
846.3 

oooo 

.400 

i.$5   .1257 

.811 

.000194 

.000213  5°54 

47H 

•463 

725.6 

000 

.372 

•945 

.1087 

.701 

.000225 

.000245  4459 

4077 

.265 

627.6 

oo 

•348 

.884 

.0951 

.6,4 

.000256 

.000280  3901 

3568 

.107 

549-6 

•324 

.823   .0824 

•S32 

.000296 

.000323  3384 

3°93 

.960 

476.1 

.762  ;.0707 

•456 

.000345 

.000377  2899 

2652 

.823 

408.1 

'•  76 

.701  1.0598 

.386 

.000408 

.000446 

2454 

244 

.696 

345-4 

L  52 

.0499 

.322 

.000489 

.000536 

2046 

871 

.58 

2880 

•  32 

.589 

.0423 

•273 

.000577 

.00063, 

1734 

586 

•49 

244.1 

.  12 

•538 

•°353 

.228 

.00069, 

.000756 

1451 

324 

203.8 

6 

•  92 

.488 

.0290 

.187 

.000842 

.000921 

"12 

086 

•33 

166.8 

1:  11 

•447 

.0243 

•*S7 

00122" 

roi10 

988 

9I2 

.28 

140.5 

9  •  44 

!s66 

.0163 

.130 

.00149 

.00164 

.190 

94.0 

10  .  28 

•325 

.0129 

.0830 

.00190 

.00208 

528 

482 

.150 

74-3 

ii  .  16 

•295 

.0!06 

.068* 

.00230 

.00252 

434 

396 

.123 

61.0 

12  .  04 

.264 

.00849 

.0548 

.00287 

.00314 

348 

.0989 

49.0 

13  .092 

•234 

.00665 

.0429 

.00367 

.00402 

273 

250 

•°774 

38.4 

14  .080 

.203 

.00503 

.0324 

.00485 

.00530 

206 

188 

.0585 

29.0 

15  .072 

16  •*' 

'III 

.00407 

.0263 

.00599 

.00657 

.00839 

167 

153 

.0474 

11 

ID 
11 

•vvt 
.056 
.048 

.103 
.142 

.122 

.00246 
00181 

.0159 
.0117 

.00752 
.0099 
•  0135 

.0108 
.0147 

74-2 

Ci 

.0287 

14.2 
IO.4 

•9 

.O4O 

.IO2 

.00126 

.00811 

.0194 

.0212 

1.6 

47  i 

!oi46 

7.26 

.036 

.0914 

00102 

.00657 

.0239 

.0262 

1.8 

.on8 

5-88 

21 

.032 

.0813 

.000804 

.00519 

.0304 

•0331 

29 

30.1 

.00936 

4.64 

22 

.028 

.0711 

.OOo6l6 

.00397 

.0396 

•0433 

53 

23.0 

.00717 

3-56 

23 

.024 

.O6l0 

.000452 

.00292 

•°539 

.0589 

8.5 

170 

.00526 

.61 

24 

.022 

•°559 

.000380 

.00245 

.0642 

.0701 

5-6 

14.3 

.00443 

.19 

25 

O2O 

.0508 

.000314 

.00203 

.0778 

.0849 

2.8 

.00366 

.80 

26 

018 

•0457 

.000254 

.00164 

.0958 

.105 

0.4 

9-54 

.00296 

•47 

27 

0164 

.0417 

.000211 

.00136 

.116 

•123 

8.65 

793 

.00246 

.22 

28 

0148 

.0376 

.000172 

.001  i  i 

.141 

7.07 

6-45 

.00200 

•893 

29 

0136 

•°345 

.000145 

.000937 

.168 

.183 

5-95 

5-45 

.00169 

•839 

0124 

•0315 

.000121 

.000779 

.202 

.221 

4.86 

4-53 

.00141 

.697 

3' 

0116 

.0295 

.000106 

.000682 

.230 

.252 

4-34 

.00123 

.610 

32 

0108 

.0274 

.0000916 

.000591 

.266 

.291 

3-75 

3-44 

.00107 

•529 

33 

OIOO 

.0254 

.0000785 

.000507 

•311 

•339 

3-22 

•94 

.000914 

34 

0092 

.0234 

.0000665 

.000429 

•367 

.402 

2.73 

•50 

.000774 

.384 

35 

0084 

.O2I3 

.0000554 

.000358 

•44° 

.481 

2.27 

.08 

.000645 

.320 

36 
37 

.0076 

.0068 

.0193 
.0173 

.0000454 
.0000363 

.000293 
.000234 

.540 
.672 

$ 

1.86 
1.49 

s 

.000548 
.000423 

.262 

.210 

38 

.0060 

.0152 

.0000283 

•944 

1.16 

.04 

.000329 

.163 

39 

.0052 

.0132  ,.  0000212 

.000137 

T-T5 

1.26 

.870 

.796 

.000247 

.123 

40 

.0048 

.0122 

.0000181 

.000117 

1.32 

1.47 

•759 

•679 

.000211 

.104 

41 

.0044 

.0112 

.0000152 

.0000981 

i.  60 

i-75 

.624 

•570 

.000177 

.0878 

42 

.0040 

.0000811 

1.94 

2.13 

•  516 

.471 

.000146 

.0726 

43 

.0036 

.00914;.  0000102 

.0000657 

2.39 

2.62 

.418 

.0588 

44 

.0032 

.00813 

.00000804 

.0000519 

3-°4 

3-32 

•330 

.301 

.0000936 

.0464 

45 

.0028 

.0071! 

.00000616 

.0000397 

3-96 

4-33 

•253 

.230 

.0000717 

.0356 

46 

.0024 

.oo6io|  .00000452 

.0000292 

5-39 

5-9° 

•185 

.170 

.0000527 

.0261 

47 

.0020 

.00508  .00000314 

.0000203 

7.76 

8-49 

.128 

.0000366 

.Ol8l 

48 

.0016 

.004061.00000201 

.0000130 

12.20 

13-3 

.0824 

•0754 

.0000234 

.OIl6 

49 

.0012 

.00305.00000113 

.00000730 

21.6 

23-5 

.0464 

.0425 

.0000132 

•00653 

5° 

.0010 

.00254  .000000785 

.00000507 

31-1 

33-9 

.0322 

.0294 

.00000914 

.00453 

296 


EXPERIMENTAL   PHYSICS. 


TABLE   XV. 

Specific  Resistances  of  Wires  of  Different  Metals  and  Alloys. 


NAME  OF  CONDUCTOR. 

SPECIFIC  RESIST- 
ANCE IN  OHMS. 

RESISTANCE  IN  OHMS. 

i  MBTER  WEIGH- 
ING x  GRAM. 

ioo  METERS  i  MM. 
IN  DIAMETER. 

Silver  (annealed)       .     .     . 

I.492XIO"6 

•JS1? 

1.899 

Silver  (hard  drawn)  .     .     . 

1.62 

.165 

2.062 

Copper  (annealed)    . 

1.584 

.1415^ 

2.017 

Copper  (hard  drawn)     .     . 

1.621 

Mi 
•1443 

2.063 

Gold  (annealed)   .... 

2.041 

.4007 

2.598 

Gold  (hard  drawn)    .     .     . 

2.077 

.4076 

2.644 

Aluminium  (annealed)   .     . 

2.889 

•0743 

3-^S- 

Zinc  (compressed)     . 

5.58 

•3995 

7-I05 

Platinum  (annealed)  . 

8.981 

1.925 

11  -435 

Iron  (annealed)    .... 

9.636 

•75** 

12.27 

Nickel  (annealed)     .     .     . 

12.356 

1.052 

15-73 

Tin  (annealed)      .... 

13-103 

•9564 

16.68 

Lead  (compressed)   .     .     . 

19.465 

2.217 

24.78 

Antimony  (compressed) 

35-21 

2-37 

44-83 

Bismuth  (compressed)  . 

130.1 

12.8 

165.60 

Mercury  (liquid*)     .     .     . 

94-34 

12.826 

120.  II 

Alloy  (2  Pt  +  i  Ag)  .     .     . 

24.187 

2.907 

30-79 

Alloy  (2  Au+  i  Ag)       .  V 

10.776 

1.638 

13.72 

Alloy  (9  Pt  +  i  Ir)    .     .     . 

21.633 

4.651 

27-54 

German  silver  

20.76 

1.817 

26.43 

*  According  to  a  very  careful  determination  by  Lord  Rayleigh  and  Mrs.  Sidgwick 
(Phil.  Trans.,  Part  I.,  1883),  a  column  of  mercury,  one  square  millimeter  in  cross- 
section,  which  at  o°  C.  has  a  resistance  of  an  ohm,  is  106.21  centimeters  in  length. 
At  an  International  Conference  at  Paris  in  1884  it  was  agreed  to  define  the  "legal 
ohm  "  as  "  the  resistance  of  a  column  of  mercury  106  centimeters  long  and  one  square 
millimeter  in  section  at  the  temperature  of  melting  ice"  The  Chamber  of  Delegates 
at  the  Chicago  Electrical  Congress  in  1 893  adopted  "  as  a  unit  of  resistance  the 
international  ohm,  which  is  based  upon  the  ohm  equal  to  IO9  units  of  resistance  of 
the  C.G.S.  system  of  electromagnetic  units,  and  is  represented  sufficiently  well  by  the 
resistance  offered  to  an  unvarying  electric  current  by  a  column  of  mercury  at  the 
temperature  of  melting  ice,  14.4521  grams  in  mass,  of  a  constant  cross-sectional  area, 
and  of  a  length  of  106.3  centimeters. 


TABLES. 


297 


TABLE   XVI. 

Conductivities  of  Pure  Metals  at  t°  C* 
Conductivity  at  o°  =  i. 


METAL. 

CONDUCTIVITY  AT  /•>  C. 

Silver      
Copper   
Gold  
Zinc                        

—  .0038278  /  +  .000009848  /2 
—  .0038701  /+  .OOOOOpOOQ  t~ 
—  .0036745  /4-  .000008443  /- 
—  .0037047  t  -\-  .000008274  t~ 

Cadmium 

—  .0036871  t  -\-  .000007575  t" 

Tin 

—  0036029  /+  .000006  136  t* 

Lead 

—  0038756  /-)-  000009  146  t2 

•\rsenic 

—  0038996  /+  0000088  79  /2 

Antimony     
Bismuth  
Iron    

—  .0030826  /-f  .000010364  /2 
—  .0035216/4-  .000005  728  /2 
—  .0051182  t+  .00001  2916  t~ 

*  From  Matthiessen. 


298 


EXPERIMENTAL   PHYSICS. 


TABLE   XVII. 

Conductivity  and  Resistance  of  Pure  Copper  at  Temperatures  from 
o°  C.  to  40°  C. 


TEMPERATURE. 

CONDUCTIVITY. 

RESISTANCE. 

TEMPERATURE. 

CONDUCTIVITY. 

RESISTANCE. 

0° 

1.  0000 

1.  0000 

21° 

.9227 

1.0838 

I 

.9961 

1.00388 

22 

.9192 

1.0879 

2 

•9923 

1.00776 

23 

.9158 

1.0920 

3 

.9885 

1.0116 

24 

.9123 

1.0961 

4 

.9847 

1.0156 

25 

.9089 

I.I003 

5 

.9809 

1.0195 

26 

•9054 

1.1044 

6 

•9771 

1.0234 

27 

.9020 

1.1085 

7 

•9734 

1.0274 

28 

.8987 

I.II27 

8 

.9696 

I-03I3 

29 

•8953 

1.1169 

9 

•9659 

"•0353 

3° 

.8920 

I.I2II 

10 

.9622 

'•0393 

31 

.8887 

I-I253 

ii 

•9585 

1-0433 

32 

.8854 

I.I295 

12 

•9549 

1-0473 

33 

.8821 

I-I337 

^3 

.9512 

l.<>5*3 

34 

.8788 

I-I379 

14 

.9476 

*-°553 

35 

.8756 

I.I42I 

'5 

.9440 

I-°593 

36 

.8723 

1.1464 

16 

.9404 

1.0634 

37 

.8691 

1.1506 

i? 

.9368 

1.0675 

38 

.8659 

I.I548 

18 

•9333 

1.0715 

39 

.8628 

I-I59I 

T9 

.9297 

1.0756 

40 

.8596 

1-1633 

20 

.9262 

1.0797 

TABLES. 


299 


TABLE   XVIII   A. 

Liquid  Resistances. 


SUBSTANCE  DISSOLVED. 

COMPOSITION. 

TEMPERATURE. 

SPECIFIC 
RESISTANCE. 

Sulphuric  acid  .... 

f  SO3HO 
'  SO3HO+  1  4  HO 
S03HO  +  1  3  HO 
LS03HO  +  499HO 

15°  c. 

19°  C. 

22°  C. 
22°  C. 

9.146  ohms 
1.336      " 
1.256      « 
I743I      " 

Sulphate  of  zinc    . 

f  ZnOSO3  +  23  HO 
ZnOS03+  24  HO 
ZnOSO3  +  105  HO 

23°  C. 
23°  C. 
23°  C. 

18.31 

1  8.02 

33.04     « 

Sulphate  of  copper    . 

CuOS03  +  45  HO 
\  CuOSO3  +  105  HO 

22°  C. 
12°  C. 

19.10       " 

31.42 

Sulphate  of  magnesia 

f  MgOSO3  +  34  HO 
\  MgOSO3  +  107  HO 

22°  C. 
22°  C. 

18.44    " 

30.06     " 

Hydrochloric  acid 

fHCl  +  7.sHO 
I  HC1  +  250  HO 

23°  C. 
23°  C. 

1.285  " 
8.177  " 

300 


EXPERIMENTAL   PHYSICS. 


TABLE   XVIII   B. 

ecific  Resistances  of  Sulphuric  Acid  at  22°  C. 
(Kohlrausck  and  Nippoldt.} 


DENSITY  OF  THE 
SOLUTION. 

PROPORTION  OF  ACID. 

SPECIFIC  RESISTANCE. 

RELATIVE  INCREASE  OF 
CONDUCTIVITY  FOR  i°C. 

.9985 

.0 

70.41 

.47-IO-- 

1.  0000 

.2 

41.05 

•47 

1.0504 

3-3 

3-252 

-653 

1.0989 

14.2 

1.787 

.646    ' 

1.1431 

20.2 

1.414 

•799 

1.2045 

28.0 

1.239 

I-31? 

1.2631 

35-2 

1.239 

1.259 

1-3163 

4i-5 

1-347 

1.410 

1-3547 

46.0 

1.487 

1.674 

1-3994 

5°-4 

1.672 

1.582 

1.4482 

55-2 

1.962 

1.417 

1.5026 

60.3 

2.412 

1.794 

TABLE   XIX. 

Internal  Resistances   {Approximate}  of  Batteries. 


ELEMENT. 

TYPE. 

RESISTANCE  IN  OHMS. 

Daniell 

Callaud 

4.00  to  5.00 

Daniell 

Meidinger 

4.00  to  9.00 

Grove 

Ordinary 

.26  to     .45 

Bunsen 

" 

.06  to     .24 

Grenet 

" 

.75  to  i.  oo 

Leclanchd 

" 

5.50  to  6.00 

Latimer-Clark 

Beetz 

I57OO.OO 

TABLES. 


30i 


TABLE   XX. 

Electromotive  Forces  of  Batteries. 


VOLTS. 


Daniell 


Daniell 


Grove 


Bunsen 


Grenet 


Leclanche" 


Volta 


Latimer-Clark 


(  Amalgamated  zinc, 
i  sulphuric  acid  -f  4  water, 
Saturated  solution  of  copper  sulphate,  f 
Copper. 

Amalgamated  zinc, 
i  sulphuric  acid  4-12  water, 
Saturated  solution  of  copper  sulphate, 
[  Copper. 

f  Amalgamated  zinc, 

I  i  sulphuric  acid  +  4  water, 

Fuming  nitric  acid, 

Platinum. 

j  Amalgamated  zinc, 
i  sulphuric  acid  +  12  water, 
Fuming  nitric  acid, 
Carbon. 


When  freshly  set  up, 

f  Amalgamated  zinc, 

-,  Solution  of  sal-ammoniac, 

I  Binoxide  of  manganese  and  carbon. 

rZinc, 

\  Ordinary  water, 

I  Copper. 

f  Zinc, 

Sulphate  of  zinc, 
j  Sulphate  of  mercury  in  paste, 

Mercury. 


1.07 
•97 

'•95 

1.94 

2.03 
1.46 

.98 
1-434 


302 


EXPERIMENTAL   PHYSICS. 


TABLE   XXI. 

Electro-chemical  Equivalents. 


ELEMENTS. 

ATOMIC 
WEIGHT. 

VALENCY. 

CHEMICAL 
EQUIVALENTS. 

ELECTRO-CHEMICAL 
EQUIVALENTS,  OR 
GRAMS  PER  COULOMB 
OF  ELECTRICITY. 

Electro-positive. 

Hydrogen    .... 

I 

I 

I 

.00001038 

Potassium    .... 

39-°3 

I 

39-°3 

.000405  I 

Sodium    

23- 

I 

23- 

.0002387 

Gold  

196.2 

3 

65-4 

.0006789 

Silver       

107.7 

i 

107.7 

.OOIIlS 

Copper  (cupric)    .     . 

63.18 

2 

3T-59 

.0003279 

Copper  (cuprous) 

63.18 

I 

63.18 

.0006558 

Mercury  (mercuric)   . 

199.8 

2 

99-9 

.001037 

Mercury  (mercurous) 

199.8 

I 

199.8 

.002074 

Tin  (stannic)    .     .     . 

117.4 

4 

29-35 

.0003046 

Tin  (stannous)      .     . 

117.4 

2 

58.7 

.0006093 

Iron  (ferric)     .     ... 

55-88 

3 

18.63 

.0001934 

Iron  (ferrous)  . 

55-88 

2 

27.94 

.OOO29OO 

Nickel 

58.6 

2 

20  •? 

.0003042 

Zinc    

64.88 

2 

*y*o 

32-44 

.0003367 

Lead  

206.4 

2 

10^.2 

.OOIO7I 

Aluminium   .... 

27.04 

3 

A  VO   *• 

9.01 

.0000935 

Electro-negative. 

Oxygen   

15.96 

2 

7.98 

.00008283 

Chlorine  .     .     .  -.-.     .- 

35-37 

I 

35-37 

.0003671 

Iodine     

126.54 

I 

126.54 

.0013134 

Bromine 

70  76 

I 

70  76 

OOO8270 

Nitrogen      .     . 

/y-  /" 

14.01 

3 

/  7     1 
4.67 

,*-/w~j  ^  y  y 
.OOOO4847 

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